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Abstract:

An optical sensor includes an optical coupler configured to receive a
first optical signal and to split the first optical signal into a second
optical signal and a third optical signal. The optical sensor further
includes a Bragg fiber in optical communication with the optical coupler.
The second optical signal and the third optical signal counterpropagate
through the Bragg fiber and return to the third port and the second port,
respectively.

Claims:

1. An optical gyroscope comprising:an optical coupler configured to
receive a first optical signal and to split the first optical signal into
a second optical signal and a third optical signal; anda Bragg fiber in
optical communication with the optical coupler such that the second
optical signal and the third optical signal counterpropagate through the
Bragg fiber and return to the optical coupler, wherein interference
between the second optical signal and the third optical signal after the
second and third optical signals have counterpropagated through the Bragg
fiber is responsive to rotations of at least a portion of the Bragg
fiber.

2. The optical gyroscope of claim 1, further comprising a light source
having an output that emits the first optical signal and is in optical
communication with the optical coupler.

5. The optical gyroscope of claim 2, wherein the light source has a
spectral distribution with a full width at half maximum of about 1
nanometer or larger.

6. The optical gyroscope of claim 2, wherein the light source has a
spectral distribution with a full width at half maximum of less than 1
nanometer.

7. The optical gyroscope of claim 1, further comprising an optical
detector in optical communication with the optical coupler to receive the
second optical signal and the third optical signal after the second and
third optical signals have counterpropagated through the Bragg fiber.

10. The optical gyroscope of claim 1, wherein the Bragg fiber is a coil.

11. The optical gyroscope of claim 10, wherein the Bragg fiber coil is
wound around a spool such that differential phase shifts due to
asymmetric variations of the temperature of the Bragg fiber coil with
respect to a mid-point of the Bragg fiber coil are reduced.

12. A method for sensing rotation, the method comprising:providing a light
signal;propagating a first portion of the light signal in a first
direction through a Bragg fiber;propagating a second portion of the light
signal in a second direction through the Bragg fiber, the second
direction opposite to the first direction;optically interfering the first
and second portions of the light signal after the first and second
portions of the light signal propagate through the Bragg fiber, thereby
producing an optical interference signal;subjecting at least a portion of
the Bragg fiber to a rotation; andmeasuring variations in the optical
interference signal caused by the rotation.

13. The method of claim 12, wherein the Bragg fiber comprises a core, the
first portion of the light signal substantially confined to the core, the
second portion of the light signal substantially confined to the core.

14. The method of claim 13, wherein the core is hollow.

15. The method of claim 14, wherein the core comprises air.

16. An optical gyroscopic system comprising:an optical coupler configured
to receive a first optical signal and to split the first optical signal
into a second optical signal and a third optical signal; andan optical
waveguide having a hollow core generally surrounded by a cladding, the
optical waveguide in optical communication with the optical coupler such
that the second optical signal and the third optical signal
counterpropagate through the optical waveguide and return to the optical
coupler, the cladding of the optical waveguide substantially confining
the counterpropagating second optical signal and third optical signal
within the hollow core, wherein interference between the second optical
signal and the third optical signal after the second and third optical
signals have counterpropagated through the optical waveguide is
responsive to rotations of at least a portion of the optical waveguide.

17. The optical gyroscopic system of claim 16, further comprising a light
source in optical communication with the optical coupler, the light
source configured to emit the first optical signal.

18. The optical gyroscopic system of claim 16, further comprising an
optical detector in optical communication with the optical coupler, the
optical detector receiving the second optical signal and the third
optical signal after having counterpropagated through the optical
waveguide.

19. A method for sensing rotation, the method comprising:providing a light
signal;propagating a first portion of the light signal in a first
direction through an optical waveguide having a hollow core generally
surrounded by a cladding;propagating a second portion of the light signal
in a second direction through the optical waveguide, the second direction
opposite to the first direction;optically interfering the first and
second portions of the light signal after the first and second portions
of the light signal propagate through the Bragg optical waveguide,
thereby producing an optical interference signal;subjecting at least a
portion of the optical waveguide to a rotation; and measuring variations
in the optical interference signal caused by the rotation.

20. A gyroscopic sensor comprising an optical waveguide having a hollow
core generally surrounded by a cladding, wherein a first optical signal
and a second optical signal counterpropagate through the optical
waveguide, the cladding of the optical waveguide substantially confining
the counterpropagating first optical signal and second optical signal
within the hollow core, wherein interference between the first optical
signal and the second optical signal is responsive to rotations of at
least a portion of the optical waveguide.

22. A perturbation sensor comprising an optical waveguide having a hollow
core generally surrounded by a cladding, wherein a first optical signal
and a second optical signal counterpropagate through the optical
waveguide, the cladding of the optical waveguide substantially confining
the counterpropagating first optical signal and second optical signal
within the hollow core, wherein interference between the first optical
signal and the second optical signal is responsive to perturbations
applied to at least a portion of the optical waveguide, wherein the
perturbations are selected from the group consisting of: magnetic fields,
electric fields, pressure, displacements, twisting, and bending applied
to at least a portion of the optical waveguide.

[0003]The present invention relates to fiber optic sensors, and more
particularly, relates to fiber optic interferometers for sensing, for
example, rotation, movement, pressure, or other stimuli.

[0004]2. Description of the Related Art

[0005]A fiber optic Sagnac interferometer is an example of a fiber optic
sensor that typically comprises a loop of optical fiber to which
lightwaves are coupled for propagation around the loop in opposite
directions. After traversing the loop, the counterpropagating waves are
combined so that they coherently interfere to form an optical output
signal. The intensity of this optical output signal varies as a function
of the relative phase of the counterpropagating waves when the waves are
combined.

[0006]Sagnac interferometers have proven particularly useful for rotation
sensing (e.g., gyroscopes). Rotation of the loop about the loop's central
axis of symmetry creates a relative phase difference between the
counterpropagating waves in accordance with the well-known Sagnac effect,
with the amount of phase difference proportional to the loop rotation
rate. The optical output signal produced by the interference of the
combined counterpropagating waves varies in power as a function of the
rotation rate of the loop. Rotation sensing is accomplished by detection
of this optical output signal.

[0007]Rotation sensing accuracies of Sagnac interferometers are affected
by spurious waves caused by Rayleigh backscattering. Rayleigh scattering
occurs in present state-of-the-art optical fibers because the small
elemental particles that make up the fiber material cause scattering of
small amounts of light. As a result of Rayleigh scattering, light is
scattered in all directions. Light that is scattered forward and within
the acceptance angle of the fiber is the forward-scattered light. Light
that is scattered backward and within the acceptance angle of the fiber
is the back-scattered light. In a fiber-optic gyroscope (FOG), both the
clockwise and the counterclockwise waves along the sensing coil (referred
to here as the primary clockwise and primary counterclockwise waves) are
scattered by Rayleigh scattering. The primary clockwise wave and the
primary counterclockwise wave are both scattered in respective forward
and backward directions. This scattered light returns to the detector and
adds noise to the primary clockwise wave and to the secondary
counterclockwise wave. The scattered light is divided into two types,
coherent and incoherent. Coherently scattered light originates from
scattering occurring along the section of fiber of length Lc
centered around the mid-point of the coil, where Lc is the coherence
length of the light source. This scattered light is coherent with the
primary wave from which it is derived and interferes coherently with the
primary wave. As a result, a sizeable amount of phase noise is produced.
Forward coherent scattering is in phase with the primary wave from which
it is scattered, so it does not add phase noise. Instead, this forward
coherent scattering adds shot noise. The scattered power is so small
compared to the primary wave power that this shot noise is negligible.
All other portions of the coil produce scattered light that is incoherent
with the primary waves. The forward propagating incoherent scattered
light adds only shot noise to the respective primary wave from which it
originates, and this additional shot noise is also negligible. The
dominant scattered noise is coherent backscattering. This coherent
backscattering noise can be large. The coherent backscattering noise has
been reduced historically by using a broadband source, which has a very
short coherence length Lc. With a broadband source, the portion of
backscattering wave originates from a very small section of fiber, namely
a length Lc of typically a few tens of microns centered on the
mid-point of the fiber coil, and it is thus dramatically reduced compared
to what it would be with a traditional narrowband laser, which has a
coherence length upward of many meters. See for example, Herve Lefevre,
The Fiber-Optic Gyroscope, Section 4.2, Artech House, Boston, London,
1993, and references cited therein.

[0008]Rotation sensing accuracies are also affected by the AC Kerr effect,
which cause phase differences between counterpropagating waves in the
interferometers. The AC Kerr effect is a well-known nonlinear optical
phenomena in which the refractive index of a substance changes when the
substance is placed in a varying electric field. In optical fibers, the
electric fields of lightwaves propagating in the optical fiber can change
the refractive index of the fiber in accordance with the Kerr effect.
Since the propagation constant of each of the waves traveling in the
fiber is a function of refractive index, the Kerr effect manifests itself
as intensity-dependent perturbations of the propagation constants. If the
power circulating in the clockwise direction in the coil is not exactly
the same as the power circulating in the counterclockwise direction in
the coil, as occurs for example if the coupling ratio of the coupler that
produces the two counterpropagating waves is not 50%, the optical Kerr
effect will generally cause the waves to propagate with different
velocities, resulting in a non-rotationally-induced phase difference
between the waves, and thereby creating a spurious signal. See, for
example, pages 101-106 of the above-cited Herve Lefevre, The Fiber Optic
Gyroscope, and references cited therein. The spurious signal is
indistinguishable from a rotationally induced signal. Fused silica
optical fibers exhibit sufficiently strong Kerr nonlinearity that for the
typical level of optical power traveling in a fiber optic gyroscope coil,
the Kerr-induced phase difference in the fiber optic rotation sensor may
be much larger than the phase difference due to the Sagnac effect at
small rotation rates.

[0009]Silica in silica-based fibers also can be affected by magnetic
fields. In particular, silica exhibits magneto-optic properties. As a
result of the magneto-optic Faraday effect in the optical fiber, a
longitudinal magnetic field of magnitude B modifies the phase of a
circularly polarized wave by an amount proportional to B. The change in
phase of the circularly polarized wave is also proportional to the Verdet
constant V of the fiber material and the length of fiber L over which the
field is applied. The sign of the phase shift depends on whether the
light is left-hand or right-hand circularly polarized. The sign also
depends on the relative direction of the magnetic field and the light
propagation. As a result, in the case of a linearly polarized light, this
effect manifests itself as a change in the orientation of the
polarization by an angle θ=VBL. This effect is non-reciprocal. For
example, in a Sagnac interferometer or in a ring interferometer where
identical circularly polarized waves counterpropagate, the magneto-optic
Faraday effect induces a phase difference equal to 2θ between the
counterpropagating waves. If a magnetic field is applied to a fiber coil,
however, the clockwise and counterclockwise waves will in general
experience a slightly different phase shift. The result is a
magnetic-field-induced relative phase shift between the clockwise and
counterclockwise propagating waves at the output of the fiber optic loop
where the waves interfere. This differential phase shift is proportional
to the Verdet constant. This phase difference also depends on the
magnitude of the magnetic field and the birefringence of the fiber in the
loop. Additionally, the phase shift depends on the orientation (i.e., the
direction) of the magnetic field with respect to the fiber optic loop as
well as on the polarizations of the clockwise and counterclockwise
propagating signals. If the magnetic field is DC, this differential phase
shift results in a DC offset in the phase bias of the Sagnac
interferometer. If the magnetic field varies over time, this phase bias
drifts, which is generally undesirable and thus not preferred.

[0010]The earth's magnetic field poses particular difficulty for Sagnac
interferometers employed in navigation. For example, as an aircraft
having a fiber optic gyroscope rotates, the relative spatial orientation
of the fiber optic loop changes with respect to the magnetic field of the
earth. As a result, the phase bias of the output of the fiber gyroscope
drifts. This magnetic field-induced drift can be substantial when the
fiber optic loop is sufficiently long, e.g., about 1000 meters. To
counter the influence of the magnetic field in inertial navigation fiber
optic gyroscopes, the fiber optic loop may be shielded from external
magnetic fields. Shielding comprising a plurality of layers of μ-metal
may be utilized.

SUMMARY OF THE INVENTION

[0011]In certain embodiments, an optical sensor is provided. The optical
sensor comprises a directional coupler comprising at least a first port,
a second port, and a third port. The first port is in optical
communication with the second port and with the third port such that a
first optical signal received by the first port is split into a second
optical signal that propagates to the second port and a third optical
signal that propagates to the third port. The optical sensor further
comprises a Bragg fiber in optical communication with the second port and
with the third port. The second optical signal and the third optical
signal counterpropagate through the Bragg fiber and return to the third
port and the second port, respectively.

[0012]In certain embodiments, a method for sensing is provided. The method
comprises providing a light signal. The method further comprises
propagating a first portion of the light signal in a first direction
through a Bragg fiber. The method further comprises propagating a second
portion of the light signal in a second direction through the Bragg
fiber, the second direction opposite to the first direction. The method
further comprises optically interfering the first and second portions of
the light signal after the first and second portions of the light signal
propagate through the Bragg fiber, thereby producing an optical
interference signal. The method further comprises subjecting at least a
portion of the Bragg fiber to a perturbation. The method further
comprises measuring variations in the optical interference signal caused
by the perturbation.

[0013]In certain embodiments, an optical system is provided. The optical
system comprises a light source having an output that emits a first
optical signal. The optical system further comprises a directional
coupler comprising at least a first port, a second port and a third port.
The first port is in optically communication with the light source to
receive the first optical signal emitted from the light source. The first
port is in optical communication with the second port and with the third
port such that the first optical signal received by the first port is
split into a second optical signal that propagates to the second port and
a third optical signal that propagates to the third port. The optical
system further comprises a Bragg fiber having a hollow core surrounded by
a cladding. The Bragg fiber is in optical communication with the second
port and with the third port such that the second optical signal and the
third optical signal counterpropagate through the Bragg fiber and return
to the third port and the second port, respectively. The cladding of the
Bragg fiber substantially confines the counterpropagating second optical
signal and third optical signal within the hollow core. The optical
system further comprises an optical detector in optical communication
with the directional coupler. The optical detector receives the
counterpropagating second optical signal and the third optical signal
after having traversed the Bragg fiber.

[0014]In certain embodiments, a perturbation sensor is provided. The
perturbation sensor comprises a Bragg fiber having a hollow core
surrounded by a cladding. A first optical signal and a second optical
signal counterpropagate through the Bragg fiber. The cladding of the
Bragg fiber substantially confines the counterpropagating first optical
signal and second optical signal within the hollow core. Interference
between the first optical signal and the second optical signal is
responsive to perturbations of at least a portion of the Bragg fiber.

BRIEF DESCRIPTION OF THE DRAWINGS

[0015]Various embodiments are described below in connection with the
accompanying drawings, in which:

[0016]FIG. 1 is a schematic drawing of an example fiber optic sensor
depicting the light source, the fiber loop, and the optical detector.

[0017]FIG. 2A is a partial perspective view of the core and a portion of
the surrounding cladding of an example hollow-core photonic-bandgap fiber
that can be used in the example fiber optic sensor.

[0018]FIG. 2B is a cross-sectional view of the example hollow-core
photonic-bandgap fiber of FIG. 2A showing more of the features in the
cladding arranged in a pattern around the hollow core.

[0019]FIG. 3 is a schematic drawing of an example Sagnac interferometer
wherein the light source comprises a narrowband light source.

[0020]FIG. 4 is a schematic drawing of an example Sagnac interferometer
driven by a narrowband light source with a modulator for modulating the
amplitude of the narrowband light source.

[0021]FIG. 5 is a schematic drawing of an example Sagnac interferometer
wherein the light source comprises a broadband light source.

[0022]FIG. 6 is a schematic drawing of an example Sagnac interferometer
driven by a broadband light source with a modulator for modulating the
amplitude of the broadband light source.

[0023]FIG. 7 schematically illustrates an experimental configuration of an
all-fiber air-core PBF gyroscope in accordance with certain embodiments
described herein.

[0024]FIG. 8A shows a scanning electron micrograph of a cross-section of
an air-core fiber.

[0025]FIG. 8B illustrates the measured transmission spectrum of an
air-core fiber and a source spectrum from a light source.

[0026]FIGS. 9A and 9B show typical oscilloscope traces of these signals,
with and without rotation, respectively.

[0027]FIG. 10 shows a trace of the f output signal recorded over one hour
while the gyroscope was at rest.

[0028]FIG. 11 schematically illustrates a cross-section of a cylindrical
fiber with a hollow core, a honeycomb inner cladding, an outer cladding,
and a jacket.

[0029]FIG. 12 illustrates the computed radial deformation as a function of
distance from fiber center for the Crystal Fibre PBF. The inset is a
magnification of the radial deformation over the inner cladding
honeycomb.

[0030]FIG. 13 shows the calculated dependence of S, Sn, and SL
on the normalized core radius for an air-core fiber (solid curves) and
for an SMF28 fiber (reference levels) at 1.5 microns.

[0031]FIGS. 14A and 14B illustrate SEM photographs of cross-sections of
two fibers compatible with certain embodiments described herein.

[0044]FIG. 24B shows the time derivative of the applied temperature change
(dashed line) of FIG. 24A with the measured resulting change in output
signal of FIG. 24A (solid line).

[0045]FIG. 25 shows the dependence of the maximum rotation rate error on
the applied temperature gradient measured in both the conventional
solid-core fiber gyroscope and in the air-core fiber gyroscope.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

[0046]A need exists to reduce or eliminate the noise and/or phase drift
induced by Rayleigh backscattering, the Kerr effect, and the
magneto-optic Faraday effect present in a fiber interferometer, as well
as other accuracy-limiting effects. In accordance with certain
embodiments disclosed herein, a hollow-core photonic-bandgap optical
fiber is incorporated in a fiber optic sensor (e.g., a Sagnac
interferometer) to improve performance or to provide other design
alternatives. While certain embodiments described herein utilize a Sagnac
interferometer, fiber optic sensors utilizing other types of
interferometers (e.g., Mach-Zehnder interferometers, Michelson
interferometers, Fabry-Perot interferometers, ring interferometers, fiber
Bragg gratings, long-period fiber Bragg gratings, and Fox-Smith
interferometers) can also have improved performance by utilizing a
hollow-core photonic-bandgap optical fiber. Fiber optic sensors utilizing
interferometry can be used to detect a variety of perturbations to at
least a portion of the optical fiber. Such perturbation sensors can be
configured to be sensitive to magnetic fields, electric fields, pressure,
displacements, rotations, twisting, bending, or other mechanical
deformations of at least a portion of the fiber.

[0047]FIG. 1 illustrates an example Sagnac interferometer 5 that comprises
a fiber optic system 12 that incorporates a photonic-bandgap fiber 13,
which, in certain embodiments, is a hollow-core photonic-bandgap fiber. A
version of a similar fiber optic system that includes a conventional
optical fiber rather than a photonic-bandgap fiber is more fully
described in U.S. Pat. No. 4,773,759 to Bergh et al., issued on Sep. 27,
1988, which is hereby incorporated herein by reference in its entirety.

[0048]The fiber optic system 12 includes various components positioned at
various locations along the fiber optic system 12 for guiding and
processing the light. Such components and their use in a Sagnac
interferometer 5 are well-known. Alternative embodiments of the system 12
having similar designs or different designs may be realized by those
skilled in the art and used in certain embodiments described herein.

[0049]As configured for the Sagnac rotation sensor 5 in FIG. 1, the fiber
optic system 12 includes a light source 16, a fiber optic loop 14 formed
with the hollow-core photonic-bandgap fiber 13 (described below in
connection with FIGS. 2A and 2B), and a photodetector 30. The wavelength
of the light output from the light source 16 may be approximately 1.50 to
1.58 microns, in a spectral region where the loss of silica-based optical
fibers is near its minimum. Other wavelengths, however, are possible, and
the wavelength of the source emission is not limited to the wavelengths
recited herein. For example, if the optical fiber comprises a material
other than silica, the wavelength is advantageously chosen in the range
of wavelengths that minimizes or reduces the loss caused by the optical
fiber. Additional detail regarding the light source of various
embodiments are described in further detail below.

[0050]The fiber loop 14 in the optic fiber system 12 in certain
embodiments advantageously comprises a plurality of turns of the
photonic-bandgap fiber 13, which is wrapped in certain embodiments about
a spool or other suitable support (not shown). By way of specific
example, the loop 14 may comprise more than a thousand turns of the
photonic-bandgap fiber 13 and may comprise a length of optical fiber 13
of about 1000 meters. The optical detector 30 may be one of a variety of
photodetectors well known in the art, although detectors yet to be
devised may be used as well.

[0051]An optional polarization controller 24 may be advantageously
included in the interferometer as illustrated in FIG. 1. The optional
inclusion of the polarization controller 24 depends on the design of the
system 12. Example polarization controllers are described, for example,
in H. C. Lefevre, Single-Mode Fibre Fractional Wave Devices and
Polarisation Controllers, Electronics Letters, Vol. 16, No. 20, Sep. 25,
1980, pages 778-780, and in U.S. Pat. No. 4,389,090 to Lefevre, issued on
Jun. 21, 1983, which are hereby incorporated by reference herein in their
entirety. The polarization controller 24 permits adjustment of the state
of polarization of the applied light. Other types of polarization
controllers may be advantageously employed.

[0052]The polarization controller 24 shown in FIG. 1 is optically coupled
to a port A of a directional coupler 26. The directional coupler 26
couples light received by port A to a port B and to a port D of the
coupler 26. A port C on the coupler 26 is optically coupled to the
photodetector 30. Light returning from the Sagnac interferometer is
received by port B and is optically coupled to port A and to port C. In
this manner, returning light received by port B is detected by the
photodetector 30 optically coupled to port C. As shown, port D terminates
non-reflectively at the point labeled "NC" "not connected"). An example
coupler that may be used for the coupler 26 is described in detail in
U.S. Pat. No. 4,536,058 to Shaw et al., issued on Aug. 20, 1985, and in
European Patent Publication No. 0 038 023, published on Oct. 21, 1981,
which are both incorporated herein by reference in their entirety. Other
types of optical couplers, however, such as fused couplers, integrated
optical couplers, and couplers comprising bulk optics may be employed as
well.

[0053]Port B of the directional coupler 26 is optically coupled to a
polarizer 32. After passing through the polarizer 32, the optical path of
the system 12 continues to a port A of a second directional coupler 34.
The coupler 34 may be of the same type as described above with respect to
the first directional coupler 26 but is not so limited, and may comprise
integrated-optic or bulk-optic devices. In certain embodiments, the light
entering port A of the coupler 34 is divided substantially equally as it
is coupled to a port B and a port D. A first portion W1 of the light
exits from port B of the coupler 34 and propagates around the loop 14 in
a clockwise direction as illustrated in FIG. 1. A second portion W2 of
the light exits from port D of the coupler 34 and propagates around the
loop 14 in a counterclockwise direction as illustrated in FIG. 1. As
shown, port C of the coupler 34 terminates non-reflectively at a point
labeled "NC." In certain embodiments, the second coupler 34 functions as
a beam-splitter to divide the applied light into the two
counterpropagating waves W1 and W2. Further, the second coupler 34 of
certain embodiments also recombines the counterpropagating waves after
they have traversed the loop 14. As noted above, other types of
beam-splitting devices may be used instead of the fiber optic directional
couplers 26, 34 depicted in FIG. 1.

[0054]The coherent backscattering noise in a fiber optic gyroscope using
an asymmetrically located phase modulator to provide bias can be
substantially reduced or eliminated by selecting the coupling ratio of
the coupler 34 to precisely equal to 50%. See, for example, J. M.
Mackintosh et al., Analysis and observation of coupling ratio dependence
of Rayleigh backscattering noise in a fiber optic gyroscope, Journal of
Lightwave Technology, Vol. 7, No. 9, September 1989, pages 1323-1328.
This technique of providing a coupling efficiency of 50% can be
advantageously used in the Sagnac interferometer 5 of FIG. 1 that
utilizes the photonic-bandgap fiber 13 in the loop 14. In a
photonic-bandgap fiber and in a Bragg fiber with a hollow (e.g.,
gas-filled) core, backscattering originates from three main sources. The
first source is bulk scattering from the gas that fills the hollow core
(as well as from the gas that fills the cladding holes), which is
negligible. The second source is bulk scattering from the solid portions
of the waveguide, namely the concentric rings of alternating index
surrounding the core in a Bragg fiber, and the solid membranes
surrounding the holes in a photonic-bandgap fiber. The third source is
surface scattering occurring at the surface of the solid portions of the
waveguide, in particular the solid that defines the outer edges of the
core, due to irregularities on this surface, or equivalently random
variations in the dimensions and shapes of these surfaces. With proper
design of the fiber, surface scattering can be minimized by reducing the
amplitude of these variations. Under such conditions, surface scattering
dominates bulk scattering, and surface scattering is much lower that of a
conventional solid-core, index-guiding single-mode fiber. The
backscattering noise in the Sagnac interferometer 5 of FIG. 1 can
therefore be reduced below the level provided by the inherently low
Rayleigh backscattering of the photonic-bandgap fiber 13. The Sagnac
interferometer 5 may be advantageously used as a fiber optic gyroscope
for high-rotation-sensitivity applications that require extremely low
overall noise.

[0055]The above-described technique of employing a coupler 34 with a
coupling efficiency of 50% works well as long as the coupling ratio of
coupler 34 remains precisely at 50%. However, as the fiber environment
changes (e.g., the coupler temperature fluctuates) or as the coupler 34
ages, the coupling ratio typically varies by small amounts. Under these
conditions, the condition for nulling the coherent backscattering
described in the previous paragraph may not be continuously satisfied.
The use of the photonic-bandgap fiber 13 in the loop 14 instead of a
conventional fiber, in conjunction with this coupling technique, relaxes
the tolerance for the coupling ratio to be exactly 50%. The
photonic-bandgap fiber 13 also reduces the backscattering noise level
arising from a given departure of the coupling ratio from its preferred
value of 50%.

[0056]A polarization controller 36 may advantageously be located between
the second directional coupler 34 and the loop 14. The polarization
controller 36 may be of a type similar to the controller 24 or it may
have a different design. The polarization controller 36 is utilized to
adjust the polarization of the waves counterpropagating through the loop
14 so that the optical output signal, formed by superposition of these
waves, has a polarization that will be efficiently passed, with minimal
optical power loss, by the polarizer 32. Thus, by utilizing both
polarization controllers, 24, 36, the polarization of the light
propagating through the fiber 12 may be adjusted for maximum optical
power. Adjusting the polarization controller 36 in this manner also
guarantees polarization reciprocity. Use of the combination of the
polarizer 32 and the polarization controllers 24, 36 is disclosed in U.S.
Pat. No. 4,773,759, cited above. See also, Chapter 3 of Herve Lefevre,
The Fiber-Optic Gyroscope, cited above.

[0057]In certain embodiments, a first phase modulator 38 is driven by an
AC generator 40 to which it is connected by a line 41. The phase
modulator 38 of certain embodiments is mounted on the optical fiber 13 in
the optical path between the fiber loop 14 and the coupler 34. As
illustrated in FIG. 1, the phase modulator 38 is located asymmetrically
in the loop 14. Thus, the modulation of the clockwise propagating wave W1
is not necessarily in phase with the modulation of the counterclockwise
propagating wave W2 because corresponding portions of the clockwise wave
W1 and the counterclockwise wave W2 pass through the phase modulator at
different times. Indeed, the modulation of the waves must be out of phase
so that the phase modulator 38 provides a means to introduce a
differential phase shift between the two waves. This differential phase
shift biases the phase of the interferometer such that the interferometer
exhibits a non-zero first-order sensitivity to a measurand (e.g., a small
rotation rate). More particularly, the modulation of the wave W1 of
certain embodiments is about 180° out of phase with the modulation
of the wave W2 so that the first-order sensitivity is maximum or about
maximum. Details regarding this modulation are discussed in U.S. Pat. No.
4,773,759, cited above.

[0058]In various embodiments, the amplitude and frequency of the phase
applied by the loop phase modulator 38 can be selected such that the
coherent backscattering noise is substantially cancelled. See, for
example, J. M. Mackintosh et al., Analysis and observation of coupling
ratio dependence of Rayleigh backscattering noise in a fiber optic
gyroscope, cited above. This selection technique can be advantageously
used in a fiber optic gyroscope utilizing a photonic-bandgap fiber loop.
The backscattering noise can thereby be reduced below the level permitted
by the inherently low Rayleigh backscattering of the photonic-bandgap
fiber, which may be useful in applications requiring extremely low
overall noise. Conversely, this technique for selecting amplitude and
frequency of the phase applied by the loop phase modulator 38 works well
as long as the amplitude and frequency of the applied phase remains
precisely equal to their respective optimum value. The use of a
photonic-bandgap fiber loop instead of a conventional fiber, in
conjunction with this technique, relaxes the tolerance on the stability
of the amplitude and frequency of the phase applied by the loop phase
modulator 38. This selection technique also reduces the backscattering
noise level that may occur when the amplitude, the frequency, or both the
amplitude and the frequency of the modulation applied by the loop phase
modulator 38 vary from their respective preferred values.

[0059]In certain embodiments, a second phase modulator 39 is mounted at
the center of the loop 14. The second phase modulator 39 is driven by a
signal generator (not shown). The second phase modulator 39 may
advantageously be utilized to reduce the effects of backscattered light,
as described, for example, in U.S. Pat. No. 4,773,759, cited above. The
second phase modulator 39 may be similar to the first phase modulator 38
described above, but the second phase modulator of certain embodiments
operates at a different frequency than the first phase modulator 38, and
the second phase modulator 39 of certain embodiments is not synchronized
with the first phase modulator 38.

[0060]In various embodiments, the photonic-bandgap fiber 13 within the
loop 14 and the phase modulators 38 and 39 advantageously comprise
polarization-preserving fiber. In such cases, the polarizer 32 may or may
not be excluded, depending on the required accuracy of the sensor. In one
embodiment, the light source 16 comprises a laser diode that outputs
linearly polarized light, and the polarization of this light is matched
to an eigenmode of the polarization maintaining fiber. In this manner,
the polarization of the light output from the laser diode 10 may be
maintained in the fiber optic system 12.

[0061]The output signal from the AC generator 40 is shown in FIG. 1 as
being supplied on a line 44 to a lock-in amplifier 46, which also is
connected via a line 48 to receive the electrical output of the
photodetector 30. The signal on line 44 to the amplifier 46 provides a
reference signal to enable the lock-in amplifier 46 to synchronously
detect the detector output signal on line 48 at the modulation frequency
of the phase modulator 38. Thus, the lock-in amplifier 46 of certain
embodiments effectively provides a band-pass filter at the fundamental
frequency of the phase modulator 38 that blocks all other harmonics of
this frequency. The power in this fundamental component of the detected
output signal is proportional, over an operating range, to the rotation
rate of the loop 14. The lock-in amplifier 46 outputs a signal, which is
proportional to the power in this fundamental component, and thus
provides a direct indication of the rotation rate, which may be visually
displayed on a display panel 47 by supplying the lock-in amplifier output
signal to the display panel 47 on a line 49. Note that in other
embodiments, the lock-in amplifier may be operated in different modes or
may be excluded altogether, and the signal can be detected by alternative
methods. See, for example, B. Y. Kim, Signal Processing Techniques,
Optical Fiber Rotation Sensing, William Burns, Editor, Academic Press,
Inc., 1994, Chapter 3, pages 81-114.

Optical Fibers

[0062]As is well known, conventional optical fibers comprise a high index
central core surrounded by a lower index cladding. Because of the index
mismatch between the core and cladding, light propagating within a range
of angles along the optical fiber core is totally internally reflected at
the core-cladding boundary and thus is guided by the fiber core.
Typically, although not always, the fiber is designed such that a
substantial portion of the light remains within the core. As described
below, the photonic-bandgap fiber 13 in the optical loop 14 also acts as
a waveguide; however, the waveguide is formed in a different manner, and
its mode properties are such that various effects that limit the
performance of a fiber interferometer that uses conventional fiber (e.g.,
a Sagnac interferometer) can be reduced by using the photonic-bandgap
fiber 13 in portions of the fiber optic system 12, particularly in the
optical loop 14.

[0064]As illustrated in FIGS. 2A and 2B, the hollow-core photonic-bandgap
fiber 13 includes a central core 112. A cladding 114 surrounds the core
112. Unlike the central core of conventional fiber, the central core 112
of the fiber 13 of certain embodiments is hollow. The open region within
the hollow core 112 may be evacuated or it may be filled with air or
other gases. The cladding 114 includes a plurality of features 116
arranged in a periodic pattern so as to create a photonic-bandgap
structure that confines light to propagation within the hollow core 112.
For example, in the example fiber 13 of FIGS. 2A and 2B, the features 116
are arranged in a plurality of concentric triangles around the hollow
core 112. The two innermost layers of holes in the example pattern are
shown in the partial perspective view of FIG. 2A. A complete pattern of
four concentric layers of holes is illustrated in the cross-sectional
view of FIG. 2B. Although the illustrated hole pattern is triangular,
other arrangements or patterns may advantageously be used. In addition,
the diameter of the core 112 and the size, shape, and spacing of the
features 116 may vary.

[0065]As illustrated by phantom lines in FIG. 2A, the features 116 may
advantageously comprise a plurality of hollow tubes 116 formed within a
matrix material 118. The hollow tubes 116 are mutually parallel and
extend along the length of the photonic-bandgap fiber 13 such that the
tubes 116 maintain the triangular grid pattern shown in FIG. 2B. The
matrix material 118 that surrounds each of the tubes 116 comprises, for
example, silica, silica-based materials or various other materials well
known in the art, as well as light-guiding materials yet to be developed
or applied to photonic-bandgap technology.

[0066]The features (e.g., holes) 116 are specifically arranged to create a
photonic-bandgap. In particular, the distance separating the features
116, the symmetry of the grid, and the size of the features 116 are
selected to create a photonic bandgap where light within a range of
frequencies will not propagate within the cladding 114 if the cladding
was infinite (i.e., in the absence of the core 112). The introduction of
the core 112, also referred to herein as a "defect," breaks the symmetry
of this original cladding structure and introduces new sets of modes in
the fiber 13. These modes in the fiber 13 have their energy guided by the
core and are likewise referred to as core modes. The array of features
(e.g., holes) 116 in certain embodiments is specifically designed so as
to produce a strong concentration of optical energy within the hollow
core 112. In certain embodiments, light propagates substantially entirely
within the hollow core 112 of the fiber 13 with very low loss. Exemplary
low loss air core photonic band-gap fiber is described in N. Venkataraman
et al., Low Loss (13 dB/km) Air Core Photonic Band-Gap Fibre, Proceedings
of the European Conference on Optical Communication, ECOC 2002.
Post-deadline Paper No. PD1.1, September 2002.

[0067]In various embodiments, the fiber parameters are further selected so
that the fiber is "single mode" (i.e., such that the core 112 supports
only the fundamental core mode). This single mode includes in fact the
two eigenpolarizations of the fundamental mode. The fiber 13 therefore
supports two modes corresponding to both eigenpolarizations. In certain
embodiments, the fiber parameters are further selected so that the fiber
is a single-polarization fiber having a core that supports and propagates
only one of the two eigenpolarizations of the fundamental core mode. In
certain embodiments, the fiber is a multi-mode fiber.

[0068]Other types of photonic-bandgap fibers or photonic-bandgap devices,
both known and yet to be devised, may be employed in the Sagnac rotation
sensors as well as interferometers employed for other purposes. For
example, one other type of photonic-bandgap fiber that may be
advantageously used is a Bragg fiber.

[0069]In accordance with certain embodiments disclosed herein, a Bragg
fiber is incorporated in a fiber optic sensor (e.g., a Sagnac
interferometer) to improve performance or to provide other design
alternatives. While certain embodiments described herein utilize a Sagnac
interferometer, fiber optic sensors utilizing other types of
interferometers (e.g., Mach-Zehnder interferometers, Michelson
interferometers, Fabry-Perot interferometers, ring interferometers, fiber
Bragg gratings, long-period fiber Bragg gratings, and Fox-Smith
interferometers) can also have improved performance by utilizing a Bragg
fiber. Fiber optic sensors utilizing interferometry can be used to detect
a variety of perturbations to at least a portion of the optical fiber.
Such perturbation sensors can be configured to be sensitive to magnetic
fields, electric fields, pressure, displacements, rotations, twisting,
bending, or other mechanical deformations of at least a portion of the
fiber.

[0070]A Bragg fiber includes a cladding surrounding a core, wherein the
core-cladding boundary comprises a plurality of thin layers of materials
with alternating high and low refractive indices. In various embodiments,
the cladding interface (i.e., the core-cladding boundary) comprises a
plurality of concentric annular layers of material surrounding the core.
The thin layers act as a Bragg reflector and contains the light in the
low-index core. For example, the core of certain embodiments is hollow
(e.g., containing a gas or combination of gases, such as air). Bragg
fibers are described, for example, in P. Yeh et al., Theory of Bragg
Fiber, Journal of Optical Society of America, Vol. 68, 1978, pages
1197-1201, U.S. Pat. No. 7,190,875, U.S. Pat. No. 6,625,364, and U.S.
Pat. No. 6,463,200, each of which is incorporated herein by reference in
its entirety. For a Bragg fiber, the amount of backscattering and
backreflection at the interface with a conventional fiber can be
different from that for other types of photonic-bandgap fibers. For
angled connections, the amount of backreflection from the interface of a
Bragg fiber and a conventional fiber can depend on the angle, wavelength,
and spatial orientation in different ways than for other types of
photonic-bandgap fibers. Furthermore, as described more fully below, in
certain embodiments, a Bragg fiber advantageously provides reduced phase
sensitivity to temperature fluctuations.

[0071]The accuracy of a fiber optic gyroscope (FOG) is generally limited
by a small number of deleterious effects that arise from undesirable
properties of the loop fiber, namely Rayleigh backscattering, the Kerr,
Faraday, and thermal (Shupe) effects. These effects induce short-term
noise and/or long-term drift in the gyroscope output, which limit the
ability to accurately measure small rotation rates over long periods of
time. The small uncorrected portions of these deleterious effects
constitute one of the main remaining obstacles to an inertial-navigation
FOG.

[0072]The use of hollow-core photonic-bandgap fiber instead of
conventional optical fiber in a Sagnac interferometer may substantially
reduce noise and error introduced by Rayleigh backscattering, the Kerr
effect, and the presence of magnetic fields. In hollow-core
photonic-bandgap fiber, the optical mode power is mostly confined to the
hollow core, which may comprise, for example, air, another gas, or
vacuum. Rayleigh backscattering as well as Kerr nonlinearity and the
Verdet constant are substantially less in air, other gases, and vacuum
than in silica, silica-based materials, and other solid optical
materials. The reduction of these effects coincides with the increased
fraction of the optical mode power contained in the hollow core of the
photonic-bandgap fiber.

[0073]The Kerr effect and the magneto-optic effect tend to induce a
long-term drift in the bias point of the Sagnac interferometer, which
results in a drift of the scale factor correlating the phase shift with
the rotation rate applied to the fiber optic gyroscope. In contrast,
Rayleigh backscattering tends to introduce mostly short-term noise in the
measured phase, thereby raising the minimum detectable rotation rate.
Each of these effects interferes with the extraction of the desired
information from the detected optical signal. The incorporation of the
hollow-core photonic-bandgap fiber 13 into the interferometer 5 in
certain embodiments advantageously diminishes these effects.

[0074]A parameter, ηnl, is defined herein as the fractional
amount of fundamental mode intensity squared in the solid portions of the
photonic-bandgap fiber. Similarly, a parameter, η, is defined herein
as the fractional amount of fundamental mode power in the solid portions
of the photonic-bandgap fiber. The phase drift caused by the Kerr
nonlinearity is proportional to the parameter ηnl, and the phase
drift caused by the magneto-optic effect, as well as the noise introduced
by Rayleigh backscattering, are each proportional to the parameter,
η, provided that η is not too small. An analysis of the effect of
ηnl is set forth below for the Kerr effect. Similar analyses can
be performed for Rayleigh backscattering and the magneto-optic Faraday
effects, using the parameter η.

Kerr Effect

[0075]Since some of the mode energy resides in the holes including the
core of the photonic-bandgap fiber and some of mode energy resides in the
solid portions of the fiber (typically a silica-based glass), the Kerr
effect in a photonic-bandgap fiber (PBF) includes two contributions. One
contribution is from the solid portions of the fiber, and one
contribution is from the holes. The residual Kerr constant of a
photonic-bandgap fiber, n2.PBF, can be expressed as the sum of these
two contributions according to the following equation:

n2.PBF=n2.solidηnl+n2.holes(1-ηnl) (1)

where n2.solid is the Kerr constant for the solid portion of the
fiber, which may comprise for example silica, and where n2.holes is
the Kerr constant for the holes, which may be, for example, evacuated,
gas-filled, or air-filled. If the holes are evacuated, the Kerr
nonlinearity is zero because the Kerr constant of vacuum is zero. With
the Kerr constant equal to zero, the second contribution corresponding to
the term n2.holes (1-ηnl) in Equation (1) is absent. In
this case, the Kerr nonlinearity is proportional to the parameter,
ηnl, as indicated by the remaining term
n2.solidηnl. However, if the holes are filled with air,
which has small but finite Kerr constant, both terms
(n2.solidηnl+n2.holes(1-ηnl)) are present.
Equation (1) above accounts for this more general case.

[0076]In certain embodiments, the parameter ηnl is equal to
Aeff/Aeff.silica, where Aeff and Aeff.silica are the
mode effective area and mode effective area in silica, respectively.
These quantities can be computed as follows:

where ng is the mode group velocity, is the relative permittivity,
and E is the electric field of the mode. Note that the parameter
ηnl is the fractional amount of fundamental mode intensity
squared in the solid portions of the photonic-bandgap fiber, not the
fractional amount of fundamental mode power in the solid portions of the
photonic-bandgap fiber, which is the regular definition of η and
which is valid for the other properties of the photonic-bandgap fiber.

[0077]For standard silica fiber, the percentage of the optical mode
contained in the cladding is generally in the range of 10% to 20%. In the
hollow-core photonic-bandgap fiber 13, the percentage of the optical mode
in the cladding 114 is estimated to be about 1% or substantially less.
Accordingly, in the photonic-bandgap fiber 13, the effective nonlinearity
due to the solid portions of the fiber may be decreased by a factor of
approximately 20. According to this estimate, by using the hollow-core
photonic-bandgap fiber 13, the Kerr effect can be reduced by at least one
order of magnitude, and can be reduced much more with suitable design.
Indeed, measurements indicate that the photonic-bandgap fibers can be
designed with a parameter ηnl small enough that the Kerr
constant of the solid portion of the fiber, n2.solid, is negligible
compared to the hole contribution, n2.holes(1-ηnl). Even in
the case where n2.solid is much larger than n2.holes, the fiber
can be designed in such a way that ηnl is sufficiently small
that n2.holes(1-ηnl) is larger than
n2.solidηnl. See, for example, D. G. Ouzounov et al.,
Dispersion and nonlinear propagation in air-core photonic-bandgap fibers,
Proceedings of the Conf. on Lasers and Electro-optics, Paper CThV5, June
2003.

Backscattering and Magneto-Optic Effects

[0078]A relationship similar to Equation (1) applies to Rayleigh
backscattering and magneto-optic Faraday effect. Accordingly, Equation
(1) can be written in the following more general form to encompass
Rayleigh backscattering and the magneto-optic Faraday effect as well as
the Kerr effect:

FPBF=Fsolidη+Fholes(1-η) (2)

In Equation (2), F corresponds to any of the respective coefficients, the
Kerr constant n2, the Verdet constant V, or the Rayleigh scattering
coefficient αs. The terms FPBF, Fsolid, and
Fholes represent the appropriate constant for the photonic-bandgap
fiber, for the solid material, and for the holes, respectively. For
example, when the Kerr constant n2 is substituted for F, Equation
(2) becomes Equation (1). When the Verdet constant V is substituted for
F, Equation (2) describes the effective Verdet constant of a
photonic-bandgap fiber.

[0079]The first term of Equation (2), Fsolidη, arises from the
contribution of the solid portion of the fiber, and the second term
Fholes(1-η) arises from the contribution of the holes. In a
conventional fiber, only the first term is present. In a photonic-bandgap
fiber, both the term for the solid portion, Fsolidη, and the
term for the hollow portion, Fholes(1-η), generally contribute.
The contributions of these terms depend on the relative percentage of
mode power in the solid, which is quantified by the parameter η. As
discussed above, if η is made sufficiently small through appropriate
fiber design, for example, the first term Fsolidη can be reduced
to a negligible value and the second term Fholes(1-η) dominates.
This is beneficial because Fholes is much smaller than Fsolid,
which means that the second term is small and thus F is small. This
second term Fholes(1-η) can be further reduced by replacing the
air in the holes with a gas having a reduced Kerr constant n2, a
reduced Verdet constant V, a reduced Rayleigh scattering coefficient
αs, or reduced values of all or some of these coefficients.
This second term Fholes(1-η) can be reduced to zero if the holes
in the fiber are evacuated.

[0080]As discussed above, the solid contributions to the Rayleigh
backscattering, the Kerr-induced phase error, and the
magnetic-field-induced phase shift on the optical signal can be decreased
by reducing the parameter η and ηnl. Accordingly, the
photonic-bandgap fiber is designed so as to reduce these parameters in
order to diminish the solid contributions of Rayleigh backscattering,
Kerr nonlinearity, and the magnetic field effects proportionally. For
example, in particular designs of the hollow-core photonic-bandgap fiber,
the value of η may be about 0.003 or lower, although this range
should not be construed as limiting. In addition to this bulk scattering
contribution, surface scattering can provide a larger contribution.

[0081]As described above, Rayleigh backscattering in an optical fiber
creates a reflected wave that propagates through the fiber in the
direction opposite the original direction of propagation of the primary
wave that produces the backscattering. Since such backscattered light is
coherent with the light comprising the counterpropagating waves W1, W2,
the backscattered light interferes with the primary waves and thereby
adds intensity noise to the signal measured by the detector 30.

[0082]Backscattering is reduced in certain embodiments by employing the
hollow-core photonic-bandgap fiber 13 in the loop 14. As described above,
the mode energy of the optical mode supported by the hollow-core
photonic-bandgap fiber 13 is substantially confined to the hollow core
112. In comparison to conventional solid-core optical fibers, less
scattering results for light propagating through vacuum, air, or gas in
the hollow core 112.

[0083]By increasing the relative amount of mode energy in the holes
(including the hollow core) and reducing the amount of mode energy in the
solid portion of the fiber, backscattering in certain embodiments is
reduced. Accordingly, by employing the photonic-bandgap fiber 13 in the
loop 14 of the fiber optic system 12, backscattering can be substantially
reduced.

[0084]A hollow-core fiber in certain embodiments also reduces the effect
of a magnetic field on the performance of the interferometer. As
discussed above, the Verdet constant is smaller in air, gases, and vacuum
than in solid optical materials such as silica-based glasses. Since a
large portion of the light in a hollow-core photonic-bandgap fiber
propagates in the hollow core, the magneto-optic-induced phase error is
reduced. Thus, less magnetic-field shielding is needed.

Light Sources

[0085]Laser light comprising a number of oscillatory modes, or
frequencies, e.g., light from a superfluorescent fiber source (SFS), may
also be used in the rotation sensing device described herein to provide a
lower rotation rate error than is possible with light from a
single-frequency source under similar conditions. Multimode lasers may
also be employed in some embodiments. In particular, the Kerr-induced
rotation rate error is inversely proportional to the number of
oscillating modes in the laser because multiple frequency components
cause the self-phase modulation and cross-phase modulation terms in the
Kerr effect to at least partially average out, thereby reducing the net
Kerr-induced phase error. A mathematical analysis of this phenomena and
examples of reductions in the Kerr-induced phase error are disclosed in
U.S. Pat. No. 4,773,759, cited above.

[0086]Although a superfluorescent light source may be used with the fiber
optic system 12 of FIG. 1, the system 12 of certain embodiments
incorporates a light source 16 that outputs light having a substantially
fixed single frequency. Because the scale factor of a fiber optic
gyroscope depends on the source mean wavelength, random variations in
this wavelength will lead to random variations in the wavelength factor,
which introduces undesirable error in the measured rotation rate. Light
sources having a substantially stable output wavelength have been
developed for telecommunications applications, and these sources are thus
available for use in fiber optic rotation sensing systems. These light
sources, however, are typically narrowband sources. Accordingly,
utilization of these narrowband stable-frequency light sources with a
conventional optical fiber would be inconsistent with the above-described
use of broadband multimode laser sources to compensate for the Kerr
effect.

[0087]However, FIG. 3 illustrates an interferometer 305 in accordance with
certain embodiments described herein that can achieve a substantially
stable wavelength while reducing the Kerr contributions to the drift in
the interferometer bias. The interferometer 305 comprises an optical
fiber system 312 that includes a stable-frequency narrowband light source
316 in combination with the hollow-core photonic-bandgap fiber 13. By
introducing the hollow-core photonic-bandgap fiber 13 into the fiber
optic system 312, the conventionally available narrowband light source
316 having a substantially stable-frequency output can be advantageously
used. The Sagnac interferometer 305 in FIG. 3 is similar to the Sagnac
interferometer 5 of FIG. 1, and like elements from FIG. 1 are identified
with like numbers in FIG. 3. As described above with respect to the fiber
optic system 12 of FIG. 1, the fiber optic system 312 of FIG. 3 also
includes an optical loop 14 that comprises a length of the hollow-core
photonic bandgap fiber 13. The narrowband light source 316 advantageously
comprises a light-emitting device 310 such as a laser or other coherent
light source. Examples of a light-emitting laser 310 include a laser
diode, a fiber laser, or a solid-state laser. In certain embodiments,
operating the FOG with a narrow-band laser diode advantageously offers
significant advantages over the current broadband sources, including a
far greater wavelength stability and thus scale-factor stability, and
possibly a lower cost. Other lasers or other types of narrowband light
sources may also be advantageously employed in other embodiments. In some
embodiments, the narrowband light source 316 outputs light having an
example FWHM spectral bandwidth of about 1 GHz or less, of about 100 MHz
or less, or about 10 MHz or less. Light sources having bandwidths outside
the cited ranges may also be included in other embodiments.

[0088]As discussed above, the light source 316 of certain embodiments
operates at a stable wavelength. The output wavelength may, for example,
not deviate more than about ±10-6 (i.e., ±1 part per million
(ppm)) in some embodiments. The wavelength instability is about
±10-7 (i.e., ±0.1 ppm) or lower in certain embodiments.
Narrowband light sources that offer such wavelength stability, such as
the lasers produced widely for telecommunication applications, are
currently available. Accordingly, as a result of the use of a
stable-wavelength light source, the stability of the Sagnac
interferometer scale factor is enhanced.

[0089]A narrowband light source will also result in a longer coherence
length in comparison with a broadband light source and will thus increase
the contribution of noise produced by coherent backscattering. For
example, if the clockwise propagating light signal W1 encounters a defect
in the loop 14, the defect may cause light from the light signal W1 to
backscatter in the counterclockwise direction. The backscattered light
will combine and interfere with light in the counterclockwise propagating
primary light signal W2. Interference will occur between the
backscattered W1 light and the counterclockwise primary light W2 if the
optical path difference traveled by these two light signals is
approximately within one coherence length of the light. For scatter
points farther away from the center of the loop 14, this optical path
difference will be largest. A larger coherence length therefore causes
scatter points farther and farther away from the center of the loop 14 to
contribute to coherent noise in the optical signal, which increases the
noise level.

[0090]In certain embodiments, a coherence length which is less than the
length of the optical path from port B of the coupler 34 to port D would
reduce the magnitude of the coherent backscatter noise. However, a
narrowband light source, such as the narrowband source 316, has a
considerably longer coherence length than a broadband light source and
thus will cause more coherent backscatter if a conventional optical fiber
is used instead of the photonic-bandgap fiber 13 in the embodiment of
FIG. 3. However, by combining the use of the stable-frequency narrowband
light source 316 with the hollow-core photonic-bandgap fiber 13 as shown
in FIG. 3, the coherent backscattering can be decreased because the
hollow-core photonic-bandgap fiber 13 reduces scattering as described
above. The bandwidth of the narrowband source 316 of certain embodiments
is selected such that the optical power circulating in either direction
through the optical loop 14 is smaller than the threshold power for
stimulated Brillouin scattering calculated for the specific fiber used in
the coil.

[0091]By employing the narrowband stable wavelength optical source 316 in
conjunction with the hollow-core photonic-bandgap fiber 13 in accordance
with FIG. 3, scale factor instability resulting from the fluctuating
source mean wavelength can be decreased while reducing the contributions
of the Kerr nonlinearities as well as coherent backscattering.

[0092]If the Kerr effect is still too large and thus introduces a
detrimental phase drift that degrades the performance of the fiber optic
system 312 of FIG. 3, other methods can also be employed to reduce the
Kerr effect. One such method is implemented in a Sagnac interferometer
405 illustrated in FIG. 4. The Sagnac interferometer 405 includes a fiber
optic system 412 and a narrowband source 416. The narrowband source 416
of FIG. 4 comprises a light-emitting device 410 in combination with an
amplitude modulator 411. The light-emitting device 410 may advantageously
be similar to or the same as the light-emitting device 310 of FIG. 3. The
optical signal from the light-emitting device 310 is modulated by the
amplitude modulator 411. In certain embodiments, the amplitude modulator
411 produces a square-wave modulation, and, in certain embodiments, the
resulting light output from the narrowband source 416 has a modulation
duty cycle of about 50%. The modulation is maintained in certain
embodiments at a sufficiently stable duty cycle. As discussed, for
example, in U.S. Pat. No. 4,773,759, cited above, and in R. A. Bergh et
al, Compensation of the Optical Kerr Effect in Fiber-Optic Gyroscopes,
Optics Letters, Vol. 7, 1992, pages 282-284, such square-wave modulation
effectively cancels the Kerr effect in a fiber-optic gyroscope.
Alternatively, as discussed, for example, in Herve Lefevre, The
Fiber-Optic Gyroscope, cited above, other modulations that produce a
modulated signal with a mean power equal to the standard deviation of the
power can also be used to cancel the Kerr effect. For example, the
intensity of the light output from the light source 416 may be modulated
by modulating the electrical current supplied to the light-emitting
device 410.

[0093]In certain embodiments, other techniques can be employed in
conjunction with the use of a narrowband light source 416 of FIG. 4, for
example, to reduce noise and bias drift. For example, frequency
components can be added to the narrowband light source 416 by frequency
or phase modulation to effectively increase the bandwidth to an extent.
If, for example, the narrowband light source 416 has a linewidth of about
100 MHz, a 10-GHz frequency modulation will increase the laser linewidth
approximately 100 times, to about 10 GHz. Although a 10-GHz modulation is
described in this example, the frequency modulation does not need to be
limited to 10 GHz, and may be higher or lower in different embodiments.
The phase noise due to Rayleigh backscattering is inversely proportional
to the square root of the laser linewidth. Accordingly, an increase in
linewidth of approximately 100 fold results in a 10-fold reduction in the
short-term noise induced by Rayleigh backscattering. Refinements in the
design of the photonic-bandgap fiber 13 to further reduce the parameter
η can also be used to reduce the noise due to Rayleigh scattering to
acceptable levels.

[0094]FIG. 5 illustrates an embodiment of a Sagnac interferometer 505 that
incorporates a broadband source 516 that may be advantageously used in
conjunction with the hollow-core photonic-bandgap fiber 13 in an optical
fiber system 512 in order to mitigate Kerr non-linearity, Rayleigh
backscattering and magnetic-field effects. Accordingly, the bias drift as
well as the short-term noise can be reduced in comparison to systems
utilizing narrowband light sources.

[0095]The broadband light source 516 advantageously comprises a broadband
light-emitting device 508 such as, for example, a broadband fiber laser
or a fluorescent light source. Fluorescent light sources include
light-emitting diodes (LEDs), which are semiconductor-based sources, and
superfluorescent fiber sources (SFS), which typically utilize a
rare-earth-doped fiber as the gain medium. An example of a broadband
fiber laser can be found in K. Liu et al., Broadband Diode-Pumped Fiber
Laser, Electron. Letters, Vol. 24, No. 14, July 1988, pages 838-840.
Erbium-doped superfluorescent fiber sources can be suitably employed as
the broadband light-emitting device 508. Several configurations of
superfluorescent fiber sources are described, for example, in Rare Earth
Doped Fiber Lasers and Amplifiers, Second Edition, M. J. F. Digonnet,
Editor, Marcel Dekker, Inc., New York, 2001, Chapter 6, and references
cited therein. This same reference and other references well-known in the
art disclose various techniques that have been developed to produce
Er-doped superfluorescent fiber sources with highly stable mean
wavelengths. Such techniques are advantageously used in various
embodiments to stabilize the scale factor of the Sagnac interferometer
505. Other broadband light sources 516 may also be used.

[0096]In certain embodiments, the broadband light source 516 outputs light
having a FWHM spectral bandwidth of, for example, at least about 1
nanometer. In other embodiments, the broadband light source 516 outputs
light having a FWHM spectral bandwidth of for example, at least about 10
nanometers. In certain embodiments, the spectral bandwidth may be more
than 30 nanometers. Light sources having bandwidths outside the described
ranges may be included in other embodiments.

[0097]In certain embodiments, the bandwidth of the broadband light source
can be reduced to relax design constraints in producing the broadband
source. Use of the hollow-core photonic-bandgap fiber 13 in the Sagnac
interferometer 505 may at least partially compensate for the increased
error resulting from reducing the number of spectral components that
would otherwise be needed to help average out the backscatter noise and
other detrimental effects. The Sagnac interferometer 505 has less noise
as a result of Kerr compensation and reduced coherent backscattering. In
certain embodiments, the fiber optic system 512 operates with enhanced
wavelength stability. The system 512 also possesses greater immunity to
the effect of magnetic fields and may therefore employ less magnetic
shielding.

[0098]The fiber optic system 512 of FIG. 5 advantageously counteracts
phase error and phase drift, and it provides a high level of noise
reduction. This enhanced accuracy may exceed requirements for current
navigational and non-navigational applications.

[0099]FIG. 6 illustrates an example Sagnac interferometer 605 in
accordance with certain embodiments described herein. The Sagnac
interferometer 605 comprises an optical fiber system 612 in combination
with a broadband light source 616. The broadband source 616
advantageously comprises a broadband light-emitting device 608 in
combination with a modulator 611. In certain embodiments, the modulator
611 modulates the power of the broadband light at a duty cycle of
approximately 50%. The modulated broadband light from the broadband
source 616 contributes to the reduction or elimination of the Kerr
effect, as discussed above.

[0100]Other advantages to employing a hollow-core photonic-bandgap fiber
are possible. For example, reduced sensitivity to radiation hardening may
be a benefit. Silica fiber will darken when exposed to high-energy
radiation, such as natural background radiation from space or the
electromagnetic pulse from a nuclear explosion. Consequently, the signal
will be attenuated. In a hollow-core photonic-bandgap fiber, a smaller
fraction of the mode energy propagates in silica and therefore
attenuation resulting from exposure to high-energy radiation is reduced.

[0101]The example Sagnac interferometers 5, 305, 405, 505 and 605
illustrated in FIGS. 1, 3, 4, 5 and 6 have been used herein to describe
the implementation and benefits of the hollow-core bandgap optical fiber
13 of FIGS. 2A and 2B to improve the performances of the interferometers.
It should be understood that the disclosed implementations are examples
only. For example, the interferometers 5, 305, 405, 505 and 605 need not
comprise a fiber optic gyroscope or other rotation-sensing device. The
structures and techniques disclosed herein are applicable to other types
of sensors or other systems using fiber Sagnac interferometers as well.

[0102]Although gyroscopes for use in inertial navigation, have been
discussed above, hollow-core photonic-bandgap fiber can be employed in
other systems, sub-systems, and sensors using a Sagnac loop. For example,
hollow-core photonic-bandgap fiber may be advantageously used in fiber
Sagnac perimeter sensors that detect movement of the fiber and intrusion
for property protection and in acoustic sensor arrays sensitive to
pressure variations applied to the fiber. Perimeter sensors are
described, for example, in M. Szustakowski et al., Recent development of
fiber optic sensors for perimeter security, Proceedings of the 35th
Annual 2001 International Carnahan Conference on Security Technology,
16-19 Oct. 2001, London, UK, pages 142-148, and references cited therein.
Sagnac fiber sensor arrays are described in G. S. Kino et al., A
Polarization-based Folded Sagnac Fiber-optic Array for Acoustic Waves,
SPIE Proceedings on Fiber Optic Sensor Technology and Applications 2001,
Vol. 4578 (SPIE, Washington, 2002), pages 336-345, and references cited
therein. Various applications described herein, however, relate to fiber
optic gyroscopes, which may be useful for navigation, to provide a range
of accuracies from low accuracy such as for missile guidance to high
accuracy such as aircraft navigation. Nevertheless, other uses, both
those well-known as well as those yet to be devised, may also benefit
from the advantages of various embodiments described herein. The specific
applications and uses are not limited to those recited herein.

[0103]Also, other designs and configurations, those both well known in the
art and those yet to be devised, may be employed in connection with the
innovative structures and methods described herein. The interferometers
5, 305, 405, 505 and 605 may advantageously include the same or different
components as described above, for example, in connection with FIGS. 1,
3, 4, 5 and 6. A few examples of such components include polarizers,
polarization controllers, splitters, couplers, phase modulators, and
lock-in amplifiers. Other devices and structures may be included as well.

[0104]In addition, the different portions of the optical fiber systems 12,
312, 412, 512 and 612 may comprise other types of waveguide structures
such as integrated optical structures comprising channel or planar
waveguides. These integrated optical structures may, for example, include
integrated-optic devices optically connected via segments of optical
fiber. Portions of the optical fiber systems 12, 312, 412, 512 and 612
may also include unguided pathways through free space. For example, the
optical fiber systems 12, 312, 412, 512 and 612 may include other types
of optical devices such as bulk-optic devices having pathways in free
space where the light is not guided as in a waveguide as well as
integrated optical structures. However, much of the optical fiber system
of certain embodiments includes optical fiber which provides a
substantially continuous optical pathway for light to travel between the
source and the detector. For example, photonic-bandgap fiber may
advantageously be used in portions of the optical fiber systems 12, 312,
412, 512 and 612 in addition to the fiber 13 in the loop 14. In certain
embodiments, the entire optical fiber system from the source to and
through the loop and back to the detector may comprise photonic-bandgap
fiber. Some or all of the devices described herein may also be fabricated
in hollow-core photonic-bandgap fibers, following procedures yet to be
devised. Alternatively, photonic-bandgap waveguides and photonic-bandgap
waveguide devices other than photonic-bandgap fiber may be employed for
certain devices.

[0105]Several techniques have been described above for lowering the level
of short-term noise and bias drift arising from coherent backscattering,
the Kerr effect, and magneto-optic Faraday effect. It is to be understood
that these techniques can be used alone or in combination with each other
in accordance with various embodiments described herein. Other techniques
not described herein may also be employed in operating the
interferometers and to improve performance. Many of these techniques are
well known in the art; however, those yet to be developed are considered
possible as well. Also, reliance on any particular scientific theory to
predict a particular result is not required. In addition, it should be
understood that the methods and structures described herein may improve
the Sagnac interferometers in other ways or may be employed for other
reasons altogether.

Temperature (Shupe) Effects

[0106]The optical phase of a signal traveling in a conventional optical
fiber is a relatively strong function of temperature. As the temperature
changes, the fiber length, radius, and refractive indices all change,
which results in a change of the signal phase. This effect is generally
sizable and detrimental in phase-sensitive fiber systems such as the
fiber sensors utilizing conventional fibers. For example, in a fiber
sensor based on a Mach-Zehnder interferometer with 1-meter long arms, a
temperature change in one of the arms as small as 0.01 degrees Celsius is
sufficient to induce a differential phase change between the two arms as
large as about 1 radian. This is about a million times larger than the
typical minimum detectable phase of an interferometric sensor (about 1
microradian). Dealing with this large phase drift is often a significant
challenge.

[0107]A particularly important fiber optic sensor where thermal effects
have been troublesome is the fiber optic gyroscope (FOG). Although the
FOG utilizes an inherently reciprocal Sagnac interferometer, even a small
asymmetric change in the temperature distribution of the Sagnac coil
fiber will result in a differential phase change between the two
counter-propagating signals, a deleterious effect known as the Shupe
effect. See, e.g., D. M. Shupe, "Thermally induced nonreciprocity in the
fiber-optic interferometer," Appl. Opt., Vol. 19, No. 5, pages 654-655
(1980); D. M. Shupe, "Fibre resonator gyroscope: sensitivity and thermal
nonreciprocity," Appl. Opt., Vol. 20, No. 2, pages 286-289 (1981).
Because the Sagnac is a common-path interferometer, the two signals see
almost the same thermally induced change and this differential phase
change is much smaller than in a Mach-Zehnder or Michelson
interferometer, but it is not small enough for high-accuracy applications
which advantageously have extreme phase stability. In this and other
fiber sensors and systems, thermal effects have been successfully fought
with clever engineering solutions. These solutions, however, generally
increase the complexity and cost of the final product, and they can also
negatively impact its reliability and lifetime.

[0108]In certain embodiments, a hollow-core fiber-optic gyroscope has
similar short-term noise as a conventional gyroscope (random walk of
about 0.015 deg/ hr) and a dramatically reduced sensitivity to Kerr
effect (by more than a factor of 50), temperature transients (by about a
factor of 6.5), and Faraday effect (by about a factor greater than 10).

[0109]The fundamental mode of a hollow-core photonic-bandgap fiber (PBF)
travels mostly in the core containing one or more gases (e.g., air),
unlike in a conventional fiber where the mode travels entirely through
silica. Since gases or combinations of gases (e.g., air) have much lower
Kerr nonlinearity and refractive index dependences on temperature than
does silica of a conventional solid-core fiber, in a hollow-core PBF
these generally deleterious effects are significantly reduced as compared
to a conventional solid-core fiber. See, e.g., D. G. Ouzounov, C. J.
Hensley, A. L. Gaeta, N. Venkataraman, M. T. Gallagher and K. W. Koch,
"Nonlinear properties of hollow-core photonic band-gap fibers," in
Conference on Lasers and Electro-Optics, Optical Society of America,
Washington, D.C., Vol. 1, pages 217-219 (2005). Since the thermal
coefficient of the refractive index dn/dT is much smaller for gases than
for silica, in a hollow-core fiber the temperature sensitivity of the
mode effective index is reduced considerably. The length of a PBF of
course still varies with temperature, which means that the phase
sensitivity will not be reduced simply in proportion to the percentage of
mode energy in silica. However, it should still be reduced significantly,
an improvement beneficial to numerous applications, especially in the FOG
where it implies a reduced Shupe effect.

[0110]This feature can be extremely advantageous in fiber sensors such as
the fiber optic gyroscope (FOG), where the Kerr and thermal (Shupe)
effects are notoriously detrimental. By replacing the conventional fiber
used in the sensing coil by a hollow-core fiber, the phase drift induced
in a FOG by these two effects should be considerably smaller. Numerical
simulations described herein predict a reduction of about 100-500 fold
for the Kerr effect, and up to about 23 fold for the thermal effect. For
the very same reason, the gyro's dependence on external magnetic fields
(Faraday effect) should be greatly reduced, by a predicted factor of
about 100-500, as described herein. Relaxing the magnitude of these three
undesirable effects should result in practice in a significant reduction
in the complexity, cost, and yield of commercial FOGs.

[0111]Furthermore, if through design and fabrication improvements Rayleigh
back-scattering in PBFs can be reduced to below that of conventional
solid-core fibers, it will also be possible to operate a hollow-core
fiber gyroscope with a narrow-band communication-grade semiconductor
source instead of the current broadband source (typically an Er-doped
superfluorescent fiber source (SFS)). Since it is difficult to stabilize
the mean wavelength of an SFS to better than 1 part-per-million, this
change would offer the additional benefit of improving the source's mean
wavelength stability by one to two orders of magnitude, and possibly
reducing the source's cost.

[0112]These benefits come at the price of an increased fiber propagation
loss (e.g., about 20 dB/km). However, this loss is manageable in
practice. It amounts to only about 4 dB for a 200-meter long coil, which
is not excessive compared to the loss of the other gyroscope components
(e.g., about 15 dB). Furthermore, the state-of-the-art loss of PBFs is
likely to decrease in the future.

Example 1

[0113]This example describes the operation of an air-core photonic-bandgap
fiber gyroscope in accordance with certain embodiments described herein.
Because the optical mode in the sensing coil travels largely through air
in an air-core photonic-bandgap fiber, which has much smaller Kerr,
Faraday, and thermal constants than silica, the air-core photonic-bandgap
fiber has far lower dependencies on power, magnetic field, and
temperature fluctuations. With a 235-meter-long fiber coil, a minimum
detectable rotation rate of about 2.7°/hour and a long-term
stability of about 2°/hour were observed, consistent with the
Rayleigh backscattering coefficient of the fiber and comparable to what
is measured with a conventional fiber. Furthermore, the Kerr effect, the
Faraday effect, and Rayleigh backcattering can be reduced by a factor of
about 100-500, and the Shupe effect by a factor of about 3-11, depending
on the fiber design.

[0114]We confirm some of these predictions with the demonstration of the
first air-core fiber gyroscope. In spite of the significantly higher loss
and scattering of existing air-core fibers compared to conventional
fibers, the sensing performance of this example is comparable to that of
a conventional FOG of similar sensing-coil length. This result
demonstrates that existing air-core fibers can readily improve the
gyroscope performance in a number of ways, for example by reducing
residual thermal drifts and relaxing the tolerance on certain components
and their stabilization.

[0115]Using an air-core fiber in the example fiber gyroscope provides a
reduction of the four deleterious effects discussed above. Referring to
Equation (1) which expresses the effective Kerr constant seen by the
fundamental core mode, the Kerr constant of air
(n2,air≈2.9×10-19 cm2/W) is about three
orders of magnitude smaller than that of silica
(n2,silica≈3.2×10-16 cm2/W). Since
ηnl<<1, the effective Kerr nonlinearity will be much
smaller in an air-core fiber than in a conventional fiber. In fact, the
third-order dispersion in a particular air-core fiber n2,PBF is
about 250 times smaller than n2,silica.

[0116]For the Faraday effect, the effective Verdet constant of the
fundamental mode VPBF can be expressed using Equation (1), but with
the constants n2 replaced by the Verdet constants of silica
(Vsilica) and air (Vair). Since Vair is much weaker (about
1600 times) than Vsilica, VPBF should be also reduced by two to
three orders of magnitudes compared to a conventional fiber.

[0117]These considerations show that in a PBF gyro, the bias drifts caused
by the Kerr and Faraday effects are reduced compared to a standard
gyroscope roughly in proportion to η, the fractional mode power in
the silica portions of the PBF. The value of η calculated for a
single-mode air-core silica PBF ranges approximately from about 0.015 to
about 0.002, depending on the core radius, air filling ratio, and signal
wavelength. Consequently, in a PBF gyroscope both the Kerr-induced and
the Faraday-induced phase drifts can be reduced by a factor of about
70-500. These are substantial improvements that should greatly relax some
of the FOG engineering requirements, such as temperature control of the
loop coupler and the amount of μ-metal shielding.

[0118]An analogous reasoning applies to Rayleigh scattering. In
state-of-the-art silica fibers, the minimum loss is limited by Rayleigh
scattering and multi-phonon coupling. In contrast, in current air-core
fibers loss is believed to be limited by coupling to surface and
radiation modes due to random dimensional fluctuations. These
fluctuations have either a technological origin (such as misalignment of
the capillary tubes or periodic core diameter variations), or from the
formation of surface capillary waves on the silica membranes of the PBF
during drawing due to surface tension. Whereas the former type of
fluctuations can be reduced with improved manufacturing techniques, the
latter has a more fundamental nature and might be more difficult to
reduce, although several approaches are possible. By reducing these
fluctuations to sufficiently low levels, the Rayleigh scattering
coefficient of air-core PBFs can reach the lower limit imposed by silica:
αPBF=ηαsilica, where αsilica is the
scattering coefficient of silica and scattering in the air portions of
the waveguide has been neglected. Thus, the lowest possible effective
scattering coefficient for an air-core fiber should be smaller than for a
conventional fiber in proportion to η, i.e., again by a factor of
70-500. As discussed below, this reduction has a large positive impact on
the fiber optic gyroscope.

[0119]In a gyroscope, the phase error due to coherence interference
between the backscattered waves and the primary waves is given by
Equation (3):

δφ = Ω 2 π ηα silica L c
( 3 ) ##EQU00002##

where Ω is the solid angle of the fundamental mode inside the fiber,
and Lc is the coherence length of the light. In a conventional
gyroscope using a standard polarization-maintaining (PM) fiber, the
external numerical aperture (NA) is typically around 0.11, so the
internal solid angle is Ω=π(NA/n)2≈0.018, where n
is the core refractive index. The mode travels entirely through silica,
so η=1, and αsilica around 1.5 microns is about -105
dB/millimeter, or about 3.2×10-8 meter-1. Using these
values in Equation (3) with light supplied by a broadband Er-doped
superfluorescent fiber source (SFS) with a coherence length of
Lc≈230 microns yields δφ≈0.15
microradians, illustrating that the backcattering noise is small compared
to the typical noise of a fiber interferometer.

[0120]The backscattering coefficient of a Crystal Fibre's AIR-10-1550
air-core fiber around 1.5 microns is about 3.5 times higher than that of
an SMF28 fiber. If the gyroscope utilizes the Crystal Fiber air-core PBF
mentioned above instead, which is representative of the current state of
the art, its effective scattering coefficient
αPBF=ηαsilica is about 3.5 times larger. This
fiber has an NA around 0.12 and a mode group index close to unity
(n≈1), so its solid angle is Ω≈0.045. The
backscattering noise provided by Equation (3) is therefore
δφ≈0.43 microradians, which is still very small. Thus,
even though backscattering is larger in current air-core fibers than in
conventional fibers, the coherence length of an Er-doped SFS can be short
enough to bring the backcattering noise down to a negligible level, and
the rotation sensitivity is approximately the same with either type of
fiber.

[0121]For air-core PBFs with ultra-low backscattering, the backscattering
noise will be reduced in proportion to η, as expressed by Equation
(3). For example, for a PBF with η=0.002 and Ω=0.045, Equation
(3) predicts δφ≈0.01 microradians. The implication is
that with such a fiber, the backscattered signal is so weak that the
gyroscope could now be operated with light of much longer coherence
length and still have a reasonably low noise. In the above example, a
coherence length as large as Lc=2.2 meters (linewidth of 95 MHz)
will still produce a noise of only 1 microradians, which is low enough
for many applications. Thus, rather than using a broadband source, a
standard semiconductor laser can be used, thereby providing significant
advantages over broadband sources, including but not limited to a far
greater wavelength stability and thus scale-factor stability, and a lower
cost, all beneficial steps for inertial-navigation FOGs.

[0122]With regard to the Shupe effect, the air-core fiber provides a
substantial reduction in thermal sensitivity. When the temperature of the
sensing coil varies asymmetrically with respect to the coil's mid-point,
the two counter-propagating signals sample the resulting thermal phase
change at different times, which results in an undesirable differential
phase shift in the gyroscope output. This effect is reduced in practical
gyros by wrapping the coil fiber in special ways, such as quadrupole
winding. However, this solution is not perfect, and residual drifts due
to time-varying temperature gradients can be observed in high-sensitivity
FOGs. When the sensing coil is made with an air-core fiber, because the
mode travels mostly in air, whose index depends much more weakly on
temperature than the index of silica, the Shupe effect is substantially
reduced. For example, in the air-core fiber used in this example, the
Shupe effect is reduced by a factor of about 3.6 times compared to a
conventional fiber. An FOG made with this fiber would therefore exhibit a
thermal drift only about 28% as large as a conventional FOG. Even greater
reductions in the Shupe constant, by up to a factor of 11, are obtained
with straightforward improvements in the fiber jacket thickness and
material. This reduction in Shupe effect advantageously translates into a
greater long-term stability and simplified packaging designs.

[0123]FIG. 7 schematically illustrates an experimental configuration of an
all-fiber air-core PBF gyroscope 705 in accordance with certain
embodiments described herein used to verify various aspects, particularly
that the noise is of comparable magnitude as in a conventional fiber
gyroscope. The light source 716 comprised a commercial Er-doped fiber
amplifier 708, which produced amplified spontaneous emission centered at
1544 nanometers with a calculated bandwidth of about 7.2 nanometers. This
light from the fiber amplifier 708 was coupled through an optical
isolator 710 and a power attenuator 712 into a 3-dB fiber coupler 730 (to
provide a reciprocal output port), then into a fiber-pigtailed
LiNbO3 integrated-optic circuit (IOC) 740 that consisted of a
polarizer 742 followed by a 3-dB splitter 744 and an electro-optic
modulator 746. The output fiber pigtails 748 of the IOC 740 were
butt-coupled to a 235-meter length of coiled HC-1550-02 air-core fiber
750 manufactured by Blaze Photonics (now Crystal Fibre A/S of Birkerod,
Denmark). This fiber 750 was quadrupolar-wound on an 8.2-centimeter
mandrel and placed on a rotation stage 752. At the butt-coupling
junctions, the ends of the air-core fiber 750 were cleaved at normal
incidence while the ends of the pigtails 748 of the IOC 740 were cleaved
at an angle to eliminate unwanted Fresnel reflections. A polarization
controller 760 was placed on one of the pigtails 748 inside the loop to
maximize the visibility of the return signal. The output signal from the
reciprocal port of the 3-dB fiber coupler 730 was detected with a
photo-detector 770. The attenuator 712 was adjusted so that the detected
output power was same in all measurements (-20 dBm). Both the fundamental
frequency (f) and the second-harmonic frequency (2f) of the detected
electrical signal were extracted with a lock-in amplifier 46 and recorded
on a computer. All results shown were obtained with a lock-in integration
time of 1 second.

[0124]FIG. 8A shows a scanning electron micrograph of a cross-section of
the air-core fiber 750. FIG. 8B illustrates the measured transmission
spectrum of the air-core fiber 750 and the source spectrum from the light
source 716. The transmission spectrum includes the coupling loss between
the air-core PBF 750 and the fiber pigtails 748 of the IOC 740. The fiber
750 is almost single moded in its transmission range (about 1490-1660
nanometers). The highest measured transmission value, near the middle of
the PBF bandgap (around 1545 nanometers) is about -10 dB. Based on the
minimum fiber loss of about 19 dB/kilometer from the manufacturer, the
235-meter length of air-core fiber 750 accounts for about 4.5 dB of the
10-dB transmission loss. The rest can be assigned to a about 2.7-dB loss
at each of the two butt junctions. The fundamental mode overlap with the
silica portions (parameter η) calculated for this fiber is a few
percent. The measured birefringence was approximately 6×10-5.
From the value of the group index for the fundamental mode of the 5-meter
standard-fiber pigtails (1.44) and of the 235-meter PBF fiber (1.04,
calculated for an ideal air-core fiber with the same core radius), the
proper frequency of the Sagnac loop was calculated to be
f0≈596 kHz. Since this calculated value is approximate, it
was also measured by observing the evolution of the 2f signal when the
gyroscope was at rest as the modulation frequency was increased up to 700
kHz. The 2f signal amplitude exhibited the frequency response of a Sagnac
interferometer, i.e., a raised sine. A fit to this dependence gave a
measured proper frequency of 592 kHz, in agreement with the calculated
value. The air-core gyroscope was modulated at f=600 kHz, with a
modulation depth of π/4, close to the optimum value for maximum
sensitivity.

[0125]The gyroscope was tested by placing it on a rotation table with a
calibrated rotation rate and measuring the amplitudes of the f and 2f
signals as a function of rotation rate. FIGS. 9A and 9B show typical
oscilloscope traces of these signals, with and without rotation,
respectively. In the absence of rotation (FIG. 9A), the output contains
only even harmonics of f (mostly 2f). When the coil is rotated (FIG. 9B),
a component at f appears. The output signal at f was measured to be
proportional to the rotation rate up to the maximum tested rotation rate
of 12°/second. The proportionality constant (of the order of 10-20
mV/° s, depending on experimental conditions) was used to
calibrate the noise level of the gyroscope in units of rotation rate. A
trace of the f output signal recorded over one hour while the gyroscope
was at rest is plotted in FIG. 10. Again, the vertical axis in this curve
was calibrated not from the knowledge of the gyroscope scale factor, but
by using the rotation table calibration. Taking the peak-to-peak noise in
this curve to be 3σ, where σ is the noise standard deviation,
these results show that 3σ≈8°/hour. Thus the minimum
detectable rotation rate (one standard deviation) is
Ωmin≈2.7°/hour, or about 1/6th of Earth rate.
The long-term drift observed in FIG. 10 is about 2°/hour. The
scale factor for this gyro, calculated from the coil diameter and fiber
length, is F=0.26 s. The short-term phase noise of the interferometer
inferred from the measured minimum detectable rotation rate is therefore
ΩminF≈3.3 microradians, which is reasonably low for an
optical interferometer.

[0126]To compare this performance to that of a conventional fiber
gyroscope, the air-core PBF coil was replaced with a coil of 200 meters
of standard PM fiber. This second coil was also quadrupole-wound, on a
mandrel 2.8-centimeters in diameter. The ends of the coil fiber were
spliced to the IOC pigtails 748. The proper frequency of this Sagnac loop
was measured to be around 500 kHz (calculated value of 513 Hz) and the
calculated scale factor was F=0.076 s. The rest of the sensor, including
the IOC 740, photodetector 770, and detection electronics, were the same
as for the air-core gyroscope described above. The minimum detectable
rotation rate of this standard gyroscope was measured to be
7°/hour (1-σ short-term noise) and its long-term drift was
3°/hour. From this minimum detectable signal and the scale factor,
a phase noise of 2.6 microradians can be inferred. This value shows the
important result that the phase noise in the air-core fiber gyroscope is
comparable to that of a standard-fiber gyroscope of comparable length
using the same configuration, detection, and electronics.

[0127]The three main contributions to the short-term noise observed in the
PM-fiber gyroscope are shot noise, d, excess noise due to the broadband
spectrum of the light source, and thermal noise in the detector. Shot
noise is typically negligible. At lower detected power, the dominant
source of noise is thermal detector noise. At higher detected power, the
dominant noise is excess noise. Under the conditions of the experimental
air-core fiber gyroscope described above, the detected power was around
10 microwatts, and the observed noise (2.6 microradians) originates
almost entirely from excess noise.

[0128]For the air-core FOG, since the detector and detected power were the
same, the shot noise contribution is also 0.4 microradians. The
backscattering noise, also calculated earlier, is about 0.4 microradians
(assuming that the Rayleigh backscattering coefficient of the air-core
fiber is 1.12×10-7 m-1, or 3.5 times stronger than that
of an SMF28 fiber). Since both test gyroscopes used the same electronics
and detected power levels, the contribution of electronic noise must be
the same as in the PM-fiber gyroscope, namely about 2 microradians.

[0129]In addition, in the air-core FOG, the backreflections at the two
butt-coupled junctions were also a possible source of noise. The two butt
junctions form a spurious Michelson interferometer that is probed with
light of coherence length much shorter than the path mismatch between the
Michelson's arms, so this interferometer only adds intensity noise. The
magnitude of this noise can be estimated using the knowledge of the power
reflection at the end of an air-core fiber, which is much weaker than for
a solid-core fiber but not zero. Such estimates show that the power
reflection at the end of in this air-core fiber is about
2×10-6. Scaling by this value, the phase noise due to these
two incoherent reflections is estimated to be roughly of the order of 1
microradians. Note that this contribution can be eliminated by angling
the ends of the air-core fiber as well.

[0130]The above calculated noise levels are consistent with the
measurements: the sum of all four contributions (0.4+0.4+2.3+1=4.1) is
comparable to the measured value of 3.3 microradians. This agreement
gives credence to the estimated values of the various noise
contributions, to the assumed value of the air-core fiber's Rayleigh
scattering coefficient, and to the conclusion that in both gyroscopes
most of the noise arises from a common electronic origin.

[0131]These measurements also allow an upper-bound value to be placed on
the Rayleigh scattering of the air-core fiber. If the observed phase
noise (3.3 microradians) is assumed to be entirely due to fiber
scattering, Equation (3) can be used to show that the fiber's Rayleigh
scattering coefficient would be as high as 6.6×10-6 m-1,
or about 200 times higher than for an SMF28 fiber. The above assignment
of the various noise sources strongly suggest that this value is
unreasonably high, and that a value of 1.12×10-7 m-1 is
much more consistent with these observations.

Example 2

[0132]This example models quantitatively the dependence of the
fundamental-mode phase on temperature in an air-core fiber, and validates
these predictions by comparing them to values measured in actual air-core
fibers. The metric cited in this example is the relative change in phase
S, referred to as the phase thermal constant, and given by Equation (4):

S = 1 φ φ T ( 4 ) ##EQU00003##

where φ is the phase accumulated by the fundamental mode through the
fiber and T is the fiber temperature. With an interferometric technique,
two air-core PBFs from different manufacturers were tested and found that
their thermal constant is in the range of 1.5 to 3.2 parts per million
(ppm) per degree Celsius, or 2.5-5.2 times lower than the measured value
of a conventional SMF28 fiber (S=7.9 ppm/° C.). Each of these
values falls within 20% of the corresponding predicted number, which
lends credence to the theoretical model and to the measurement
calibration. This study shows that the reason for this reduction is due
to a drastic reduction in the dependence of the mode effective index on
temperature. The residual value of the thermal constant arises from
length expansion of the fiber, which is only marginally reduced in an
air-core fiber. Modeling shows that with fiber jacket improvements, this
contribution can be further reduced by a factor of about 2. Even without
this further improvement, the phase thermal constant of current air-core
fiber is as much as about 5 times smaller than in a conventional fiber,
which can bring forth a significant improvement in the FOG and other
phase-sensitive systems.

[0133]The phase thermal constants S of an air-guided photonic-bandgap
fiber and a conventional index-guided fiber are quantified in the
following theoretical model. The total phase φ accumulated by the
fundamental mode as it propagates through a fiber of length L is
expressed by Equation (5):

φ = 2 π n eff L λ ( 5 )
##EQU00004##

where L is the fiber length, neff the mode effective index, and the
wavelength of the signal in vacuum. Inserting Equation (5) into Equation
(4) yields the expression for the phase sensitivity per unit length and
per degree of temperature change of the fiber, as expressed by Equation
(6):

where dφ/dT is the derivative of the phase delay with respect to
temperature. S is the sum of two terms: the relative variation in fiber
length per degree of temperature change (hereafter called SL), and
the relative variation in the mode effective index per degree of
temperature change (hereafter called Sn).

[0134]If the temperature change from equilibrium is ΔT(t,l) at time
t and in an element of fiber length dl located a distance l from one end
of the fiber, the total phase change in a length L of the fiber is
expressed by Equation (7):

Δφ = 2 π n eff λ S ∫ 0 L
Δ T ( t - l / v , l ) l ( 7 )
##EQU00006##

where v=c/neff and c is the velocity of light in vacuum. Equation (7)
shows that S is a relevant parameter to characterize the phase
sensitivity to temperature in, for instance, a Mach-Zehnder
interferometer, since the total phase change is proportional to S.

[0135]Similar expressions apply to other interferometers. For example, for
a Sagnac interferometer the corresponding phase change is given by
Equation (8):

As expected, Δφ is proportional to S, and S is again the
relevant metric. The temperature sensitivity of a Sagnac interferometer,
expressed by Equation (8), can be reduced by minimizing the integral
through proper fiber winding, and/or by designing the fiber structure to
minimize S.

[0136]Because the thermal expansion coefficient of the fiber jacket
(usually a polymer) is typically two orders of magnitude larger than that
of silica, expansion of the jacket stretches the fiber, and the fiber
length change caused by jacket expansion is the dominant contribution to
SL. The index term Sn is the sum of three effects. The first
one is the transverse thermal expansion of the fiber, which modifies the
core radius and the photonic-crystal dimensions, and thus the mode
effective index. The second effect is the strains that develop in the
fiber as a result of thermal expansion; these strains alter the effective
index through the elasto-optic effect. The third effect is the change in
material indices induced by the fiber temperature change (thermo-optic
effect).

[0137]To determine Sn and SL, the thermo-mechanical properties
of the fiber are modeled by assuming that the fiber temperature is
changed uniformly from T0 to T0+dT, and calculating the fiber
length and the effective index of the fundamental mode at both
temperatures, from which, with Equation (6), Sn, SL, and S can
be calculated. FIG. 11 schematically illustrates a fiber 800 in
accordance with certain embodiments described herein. The fiber 800 is
assumed to have cylindrical symmetry and all its properties are assumed
to be invariant along its length, so S is computed in a cylindrical
coordinate system. As shown in FIG. 11, the fiber 800 is modeled as a
structure with multiple circular layers: a core 810 (e.g., doped silica
in a conventional fiber, hollow in a PBF, and shown as filled with air in
FIG. 11) of radius a0, an inner cladding 820 (e.g., silica in a
conventional fiber, a silica-air honeycomb in a PBF) generally
surrounding the core 810, an outer cladding 830 (generally pure silica)
of radius aM and generally surrounding the inner cladding 820, and a
jacket 840 (often an acrylate) generally surrounding the outer cladding
830. Each layer is assumed to remain in contact and in mechanical
equilibrium with the neighboring layers, i.e., the radial stress and the
radial deformation are continuous across fiber layer boundaries. In
certain embodiments, the core 810 has an outer diameter in a range
between about 9 microns and about 12 microns. In certain embodiments, the
inner cladding 820 has an outer diameter in a range between about 65
microns and about 72 microns. In certain embodiments, the outer cladding
830 has an outer diameter in a range between about 110 microns and about
200 microns. In certain embodiments, the jacket 840 has an outer diameter
in a range between about 200 microns and about 300 microns.

[0138]Each layer is characterized by a certain elastic modulus E,
Poisson's ratio v, and thermal expansion coefficient α. The
photonic-crystal cladding 820 is an exception in that it is not a
homogeneous material but it behaves mechanically like a honeycomb. The
implications are that (1) in a transverse direction the honeycomb can be
squeezed much more easily than a solid, which means that it has a high
transverse Poisson's ratio, and (2) in the longitudinal direction, it
behaves like membranes of silica with a total area (1-η)Ah,
where Ah is the total cross section of the honeycomb. The elastic
modulus and Poisson's ratio of a honeycomb are thus function of the air
filling ratio η. For an hexagonal pattern of air holes in silica,
they are given by Equation (9):

where ET and EL are the transverse and longitudinal Young's
modulus of the silica-air honeycomb, respectively, E0 is the Young's
modulus of silica, vT and vL are the transverse and
longitudinal Poisson's ratios of the honeycomb material, respectively,
and v0 is the Poisson's ratio of silica. The values used for these
parameters in the simulations presented here were calculated from
Equation (9) and are listed in Table 1. Comparison to a simpler model in
which the inner cladding is approximated by solid silica indicates that
the effect of the honeycomb is to increase SL by about 10-30%. The
reason is that in a honeycomb offers a lower resistance to the pull
exerted by the higher thermal expansion jacket than solid silica, thus
the fiber length expansion is increased (larger SL). The effect of
the honeycomb is thus small but not negligible. Table 1 also lists the
values of the parameters used in the simulations for the other fiber
layers.

[0139]The local deformation vector u(r) at the point r=[r, θ, z] is
given by Equation (10):

u(r)=[ur(r)0uz(z)] (10)

Only the diagonal components of the strain tensor c are non-zero:

= [ rr = ∂ u r ∂ r
θθ = u r r zz = ∂ u z
∂ z ] ( 11 ) ##EQU00009##

[0140]Hooke's law is used to relate the stress tensor σ and strain
tensor .di-elect cons. along with the effect of a temperature change
ΔT:

.di-elect cons.=s:σ+αΔT (12)

where s is the fourth-order compliance tensor, α is the thermal
expansion tensor, which also only has diagonal terms, and: denotes the
tensor product.

[0141]The deformation field uz does not vary with r and for a long
fiber, it varies linearly with z, such that it is of the form:

uz(z)=Cz (13)

where C is a constant and the z origin is chosen in the middle of the
fiber. Because uz(z) is continuous at each interface between layers,
C has the same value for all layers. Since the temperature is assumed
uniform across the fiber, Equations (12) and (13) imply that .di-elect
cons.zz and σzz are independent of r and only functions
of z. Furthermore, ur satisfies the admissibility condition:

∂ 2 u r ∂ r 2 + 1 r
∂ u r ∂ r - u r r 2 = 0 ( 14
) ##EQU00010##

whose solution is:

u r ( r ) = Ar + B r ( 15 ) ##EQU00011##

where A and B are constants specific to each layer. The coefficients A, B,
and C are solved for by imposing the following boundary conditions and
making use of Hooke's law: (i) continuity of ur(r) across all inner
layer boundaries; (ii) continuity of σrr(r) across all inner
layer boundaries; (iii) σrr=0 at r=a0, where a0 is
the fiber core radius; (iv) σrr(r)=0 at r=aM, where
aM is the outer radius of the fiber; and (v) mechanical equilibrium
on the fiber end faces, which imposes:

∫a0aM∫02πσzz(r,.thet-
a.,z=±L/2)drdθ=0 (16)

A matrix method can be used to determine A, B, and C, and thus ur(r)
and ur(z). Equation (11) then yields the strains, including
.di-elect cons.zz=SLΔT.

[0142]To illustrate the kinds of predictions this model provides, FIG. 12
shows the radial deformation as a function of distance from the fiber
center calculated for the Crystal Fibre PBF with the physical parameters
listed in Table 2. Over the honeycomb structure (inner radius of about 5
microns and outer radius of about 33.5 microns) and the silica outer
cladding (inner radius of about 33.5 microns and outer radius of about
92.5 microns), the radial deformation remains small compared to the
deformation of the acrylate jacket (inner radius of about 92.5 microns
and outer radius of about 135 microns), which is consistent with the
differences in the thermal expansion coefficient and stiffness of the
materials. The low-thermal expansion and stiff silica experiences a much
weaker deformation than the high-thermal expansion and soft acrylate.
Since the radial strain is the derivative of the radial deformation, the
inner cladding honeycomb is under compressive strain, and it relaxes the
strain over the structure by absorbing some of the deformation (the
radial deformation decreases by a factor of 4 over the honeycomb
structure) due to its very small transverse Young modulus.

[0143]Once the strain distributions are known, computation of Sn is
straightforward. In a first step, from the radial strain distribution,
the change in the dimension of each layer across the fiber cross-section
is calculated. In a second step, from the total strain distribution, the
change in refractive indices of each layer due to the elasto-optic effect
is calculated. In a third step, the change in material indices induced by
the temperature change (thermo-optic effect), which is independent of the
strains and the easiest to evaluate, is calculated. These three
contributions (change in index profile, core radius, and materials'
indices) are then combined to obtain the refractive index profile of the
fiber at T=T0+ΔT. This new profile is then imported into an
appropriate code to calculate various optical properties of the structure
(see, e.g., V. Dangui, M. J. F. Digonnet and G. S. Kino, "A fast and
accurate numerical tool to model the mode properties of photonic-bandgap
fibers," Optical Fiber Conference Technical Digest (2005)), such as the
effective index of the fundamental mode at this temperature. The code is
also used to compute the mode effective index of the unperturbed fiber,
i.e., at temperature T0. These two values of the effective index are
used in Equation (6) to compute Sn. This calculation assumes that
all parameters change linearly with temperature, which is reasonable for
small temperature excursions.

[0144]FIG. 13 shows the dependence of SL, Sn, and S on the core
radius R predicted by this model for a fiber with a cladding air-hole
radius ρ=0.495Λ, an outer cladding radius of 92.5 μm, and
an acrylate jacket of thickness 42.5 μm (parameters of the Crystal
Fibre PBF). The signal wavelength was λ=0.5Λ, close to the
middle of the fiber bandgap, with a center-to-center distance between
adjacent hollow regions in the inner cladding of Λ=3 μm. The
values of S, SL, and Sn calculated for an SMF28 fiber (see
parameters in Table 2) at λ=1.5 μm are also indicated for
comparison. SL is almost independent of core radius and is the
dominant term. The situation is reversed from a conventional SMF28 fiber,
for which Sn is significantly larger than SL. Note also that
SL is sensibly the same for the air-core and the SMF28 fibers. The
physical reason is that SL, quantifies linear expansion of the
fiber, which is similar in both fibers. SL is actually a little
lower for the air-core fiber because of the increased relative area of
silica in the outer cladding compared to the acrylate in the jacket for
this particular fiber. Therefore, the PBF has a lower overall thermal
expansion than the SMF28 fiber. For the air-core fiber, the index term
Sn generally decreases slowly with increasing core radius, except
for prominent local peaks in the ranges of 1.1Λ-1.25Λ and
1.45Λ-1.65Λ, where Sn and S increase by as much as a
factor of two. These ranges coincide precisely with the regions when
surface modes occur (highlighted in gray in FIG. 13). The reason is that
for the core radii that support surface modes, a significantly larger
fraction of the fundamental mode energy is contained in the dielectric
portions of the fiber, and the phase is more sensitive to temperature.
This result points out yet another reason why surface modes should be
avoided. Outside of these surface-mode regions, the total phase thermal
constant S=SL+Sn varies weakly with core radius. The lowest S
value in the single-mode range (R<˜1.1Λ) occurs for
R≈1.05Λ and is equal to about 1.68 ppm/° C., which
is 4.9 times smaller than for an SMF28 fiber.

[0145]Since in an air-core fiber most of the contribution to S comes from
the length term SL, the more complex index term Sn can be
neglected and it is worth developing a simple model to evaluate SL
(and thus S). The value of SL can be approximated to a good
accuracy, while gaining some physical insight for the effects of the
various parameters, by ignoring the radial terms and utilizing the
condition that the total force exerted on the fiber in the z direction is
zero. Using the notation in FIG. 11, this total force can be expressed
as:

σzz.hAh+σzz.clAcl+σzz.JAJ=-
0 (17)

where the subscripts h, cl, and J stand for honeycomb, outer cladding, and
jacket, respectively. The corresponding term for the air core is zero and
is thus absent from Equation (17). Substituting Equation (12) into
Equation (17) while neglecting the transverse terms, which are small
because the fiber radius is small compared to the fiber length, the
following approximate expression is obtained for SL:

[0146]As can be seen from Table 1, the jacket expansion term
AJEJαJ and the outer cladding expansion term
AclEclαcl are comparable in size and much larger
than the honeycomb term AhEhαh, which can be
neglected. In the denominator, the main term is the restoring force
AclEcl due to the outer cladding, which is much larger than the
force from the jacket or the honeycomb. Hence, Equation (16) can be well
approximated by:

This simple expression shows that SL can be lowered by making the
area of the outer cladding Acl as large as possible relative to the
jacket area AJ, and by using a jacket material with a low thermal
expansion. This approximate model turns out to be quite accurate. In
certain embodiments described herein, the area of the outer cladding
Acl, the area of the jacket AJ, the Young's modulus of the
outer cladding Ecl, the Young's modulus of the jacket EJ, and
the coefficient of thermal expansion of the outer cladding
αcl, and the coefficient of thermal expansion of the jacket
αJ, are selected such that the quantity

A J A cl E J E cl α J α cl ##EQU00014##

is less than or equal to 2.5, while in certain other embodiments, this
quantity is less than 1.

[0147]The parameter S was measured for two PBF fibers, namely the
AIR-10-1550 fiber manufactured by Crystal Fibre A/S and the HC-1550-02
fiber from Blaze Photonics (now Crystal Fibre A/S). SEM photographs of
the fibers' cross-sections are shown in FIGS. 14A and 14B, respectively.
Measurements were carried out using the conventional Michelson fiber
interferometer schematically illustrated in FIG. 15A. The signal source
was a 1546-nm DFB laser with a linewidth of a few MHz. The air-core fiber
was either spliced (Blaze Photonics fiber) or butt-coupled (Crystal Fiber
fiber) to one of the ports of a 3-dB coupler (SMF28 fiber) to form the
"sensing" arm of the interferometer. The far end of the PBF was similarly
coupled to a fiber-pigtailed Faraday rotator mirror (FRM) to reflect the
signal back through the fiber and thus eliminating polarization
fluctuations in the return signal due to variations in the fiber
birefringence. Most of the PBF was attached to an aluminum block placed
on a heating plate, and the fiber/block assembly was covered with a
styrofoam thermal shield (shown schematically in FIG. 15A) to maintain
the fiber temperature as uniform as possible and to reduce temperature
fluctuations due to air currents in the room. The temperature just above
the surface of the block was measured with a thermocouple (e.g., output 1
mV/° C.).

[0148]The second (reference) arm of the interferometer consisted of a
shorter length of SMF28 fiber splice to a second FRM. Together with the
non-PBF portion of the sensing arm, this entire arm was placed in a
second thermal shield (shown schematically in FIG. 15A), mostly to reduce
the amount of heating by the nearby heater of both the reference fiber
and the non-PBF portion of the sensing arm. With this arrangement, when
the heater was turned on the PBF was the only portion of the
interferometer that was significantly heated.

[0149]To measure S, the temperature of the PBF was raised to around
70° C., then the heater was turned off and as the PBF temperature
slowly dropped, both the output power of the interferometer and the fiber
temperature were measured over time and recorded in a computer. During
the measurement time window (typically tens of minutes), the phase in the
PBF arm decreased and passed many times (e.g., 50-200) through 2π, so
that the power at the interferometer output exhibited many fringes, as
illustrated in the typical experimental curves of FIG. 16. The phase
thermal constant S was calculated from the measured number of fringes
occurring in a given time interval using:

where L is the length of fiber under test, ΔT is the temperature
change occurring during the measurement interval, and Nfringes is
the number of fringes, which are illustrated in FIGS. 16A and 16B.

[0150]This approximation is justified because the temperature drop was
slow enough that the PBF temperature was uniform at all times, yet fast
enough that random phase variations in the rest of the interferometer
were negligible compared to the phase variations in the PBF. To verify
this last point, the inherent temperature stability of the interferometer
was measured by disconnecting the PBF and reconnecting the fiber ends of
the sensing arm with a short length of SMF28 fiber, as illustrated in
FIG. 15B. In a first stability test, the interferometer output was
recorded while the entire interferometer temperature was at equilibrium
room temperature. Over a period of about 30 minutes, the enclosure
temperature was found to vary by ±1° C. and the output power
varied by about one fringe only. This test showed that the interferometer
was more than stable enough to measure phase shifts of tens of fringes.

[0151]In a second test, the PBF enclosure was heated to around 70°
C., then the heater was turned off and the interferometer output was
recorded as the heater slowly cooled down. This time a larger number of
fringes were observed, which indicated that a little heat from the heater
reached through the interferometer shield and induced a differential
temperature change in the two arms. The output power varied by about 12
fringes while the enclosure temperature dropped about 18° C.
Consequently, when measuring S with the setup of FIG. 15A, residual
heating of the non-PBF portion of the interferometer introduces an error
of about 12 fringes. For this error to be small compared to the fringe
count due to the change in the PBF temperature, this fringe count should
be much larger than the error, e.g., 100 or more. This condition was met
by using a sufficiently long PBF. For the value of S≈2
ppm/° C. predicted for a PBF (e.g., FIG. 13), Equation (18)
predicts that the length required to obtain 100 fringes of phase shift
for a ΔT of 18° C. is L≈1 meter. The length of PBF
used in the measurements described herein was therefore of this order
(about 2 meters, as shown in Table 2).

[0152]As a point of comparison, the thermal constant of a conventional
solid-core fiber was measured by replacing the PBF in the experimental
setup of FIG. 15A by a 210-cm length of SMF28 fiber. The measured value
was S=7.9 ppm/° C., in excellent agreement with the value of 8.2
ppm/° C. predicted by the model using the parameter values of
Tables 1 and 2. This value is the sum of SL=2.3 ppm/° C. and
Sn=5.9 ppm/° C., i.e., the index contribution is 2.6 times
larger than the length expansion contribution. These values are
summarized in Table 3. The close agreement between measured and
calculated values gives credence to both the model and the interferometer
calibration.

[0153]The value of S was then measured for the two air-core PBFs. A
typical experimental result is shown in FIGS. 16A and 16B. The value of S
inferred for each fiber from such measurement and Equation (18) is listed
in Table 3, along with the calculated values of S, Sn, and SL.
The S values measured for the two PBFs are fairly similar, in the range
of 1.5 to 2.2 ppm/° C. As predicted, the air-core fiber guidance
mechanism results in a sizable decrease in the sensitivity of the phase
delay on temperature. This reduction is as much as a mean factor of 5.26
(measured) or 5.79 (predicted) for the Crystal Fibre PBF. The
corresponding figures for the Blaze Photonics fiber are 3.6 (measured)
and 3.14 (predicted). Again, the theoretical and measured values agree
well. The Crystal Fibre fiber exhibits a lower thermal expansion
contribution than the Blaze Photonics fiber because it has a larger area
of silica cladding relative to the jacket area, as expressed by Equation
(17). These reductions in S result mostly from a decrease in Sn by a
factor of about 100, as well as a 15%-45% reduction in SL; as
predicted by theory, in a PBF S is determined overwhelmingly by SL,
which depends only on the change in fiber length. The conclusion is that
current air-core fibers are substantially less temperature sensitivity
that conventional fibers, by a factor large enough (e.g., 3.6-5.3) that
it will translate into a significant stability improvement in fiber
sensors and other phase-sensitive fiber systems.

[0154]Even smaller values of S can be obtained with improved PBF designs.
Since in a PBF, SL is the main contribution to S, to further reduce
S, the value of SL can be reduced. This term arises from the thermal
change in the fiber length, which is driven by both the thermal expansion
coefficient and the stiffness of (i) the honeycomb cladding (e.g., silica
and air), (ii) the outer cladding (e.g., silica), and (iii) the jacket
(e.g., a polymer). Because polymers have a much higher thermal expansion
coefficient than silica, as the temperature is increased the jacket
expands more than the fiber, and thus it pulls on the fiber and increases
its length more than if the fiber was unjacketed. The jacket is therefore
generally the dominant contribution to SL. Consequently, a thinner
jacket will result in a smaller SL, the lowest value being achieved
for an unjacketed fiber. Furthermore, everything else being the same a
softer jacket (lower Young modulus) will stretch the fiber less
effectively and thus yield a lower SL. In addition, increasing the
outer cladding thickness increases the overall stiffness of the fiber
structure, thus reducing the expansion of the honeycomb and reducing
SL.

[0155]These predictions were confirmed by simulating the Blaze Photonics
fiber for various acrylate jacket thicknesses and air filling ratios, as
shown in FIG. 17. As the jacket thickness is reduced, SL decreases.
In the limit of zero jacket thickness (bare fiber), SL reaches its
lowest limit, set by the thermal expansion of the silica cladding. For
higher air filling ratios, SL is observed to be larger. The reason
is that the honeycomb then contains less silica, the fiber has a lower
overall stiffness, and the jacket expansion is less restrained by the
glass structure, resulting in a larger SL value.

[0156]The effect of the jacket material stiffness can be seen in FIG. 18,
where the calculated values of SL is graphed (for the same PBF, with
an air filling ratio of 90%) for a few standard jacket materials (metals,
polymers, and amorphous carbon covered with polyimide). To simulate
actual fiber jackets, the jacket thickness was taken to be 5 or 50
microns for polyimide (as specified), 20 microns for metals, and 20
nanometers for amorphous carbon (covered with either 2.5 or 5 microns of
polyimide). The reference jacket of the actual manufactured PBF was 50
microns of acrylate. All metal jackets yield a larger SL than did
the reference acrylate jacket (2.57 ppm/° C.). The explanation is
that while metals have a lower thermal expansion than acrylate (by about
one order of magnitude), their Young modulus is much larger than that of
both acrylate (by 2-3 orders of magnitude) and silica (by a factor of up
to 3). The silica structure is therefore pulled more effectively by the
expanding metal coating than by the acrylate jacket, and SL is
larger. Several jacket materials, however, perform better than acrylate.
A thin (20 nanometer) amorphous carbon coating with a 2.5-micron
polyimide jacket over it gives the lowest value, SL=0.67
ppm/° C. (74% reduction), followed by a 5-micron polyimide jacket
(SL=0.77 ppm/° C., 70% reduction). This is close to the
theoretical limit for a silica fiber, which is set by the thermal
expansion coefficient of silica and is equal to SL=0.55 ppm/°
C. The polyimide jacket provides the lowest value of SL because it
is much thinner than an acrylate jacket. For equal thickness, acrylate
actually performs better than polyimide. But because polyimide is a
better water-vapor barrier than acrylate, a polyimide jacket only a few
microns thick is sufficient to effectively protect the fiber against
moisture, which is not true for acrylate. The conclusion is that acrylate
unfortunately happens not to be the best choice of jacket material for
thermal performance. By coating the PBF with the above carbon-polyimide
jacket instead, an SL of only 0.67 ppm/° C., i.e., an S as
low as 0.72 ppm/° C., is attainable, which is about 11 times lower
than for a conventional fiber.

[0157]The effect of the silica outer cladding was also studied by
simulating the same PBF for increasing cladding thicknesses, assuming a
fixed 50-micron acrylate jacket and a 90% air-filling ratio. The result
is plotted as the solid curve in FIG. 19. As the outer cladding thickness
is increased, SL drops, because a thicker silica cladding better
resists the length increase of the acrylate jacket. This effect is fairly
substantial. For example, by doubling the outer cladding thickness from
the 50-micron value of the Blaze Photonics fiber to 100 microns, SL
is reduced by 55%. In the opposite limit of no outer cladding (zero
thickness), SL jumps up to more than 20 ppm/° C. The high
thermal expansion jacket is then pulling only on the silica honeycomb
structure, which has a lower Young modulus due to the air holes and thus
offers less resistance to stretching. Using a thick outer cladding is
therefore an advantageous way of reducing the thermal sensitivity of an
air-core fiber. The downside is that the fiber is then stiffer and can
therefore not be wound as tightly, which is a disadvantage in some
applications.

[0158]The dashed curve in FIG. 19 was generated using the approximate
model described herein. This curve is in very good agreement with the
exact result. Since again SL accounts for more than 90% of the
thermal constant S, this very simple model is a reliable tool to predict
the thermal constant of any fiber structure.

[0159]Thus, it is possible to reduce the thermal constant below the low
value already demonstrated in existing air-core fibers by using (i) a
jacket as thin as possible; (ii) a soft jacket material; (iii) a large
outer cladding; and/or (iv) a small air filling ratio (inasmuch as
possible). Jacket materials that satisfy (i) and (ii) include, but are
not limited to, polyimide and amorphous carbon covered by a thin layer of
polyimide. With a 5-micron polyimide jacket, the Blaze Photonics fiber
has a thermal constant of S≈0.82 ppm/° C., which is about
3.2 times smaller than in the current fiber.

[0160]In certain embodiments, the phase thermal constant S less than 8
parts-per-million per degree Celsius. In certain embodiments, the phase
thermal constant S less than 6 parts-per-million per degree Celsius. In
certain embodiments, the phase thermal constant S less than 4
parts-per-million per degree Celsius. In certain embodiments, the phase
thermal constant S less than 1.4 parts-per-million per degree Celsius. In
certain embodiments, the phase thermal constant S less than 1
part-per-million per degree Celsius.

[0161]The theoretical thermal phase sensitivity to temperature was also
calculated for a Bragg fiber with a core radius of 2 microns, surrounded
by 40 air-silica Bragg reflectors with thicknesses of 0.48 microns
(silica) and 0.72 micron (air), with an acrylate jacket thickness of 62.5
microns. This fiber exhibits a fundamental mode confined in its air core
with a radius of 1.5 microns. As shown in Table 3, these results yielded:
SL=1.15 ppm/° C., Sn=0.30 ppm/° C., and S=1.45
ppm/° C. Because the fundamental mode in a Bragg fiber travels
mostly in air, this value of S is much lower than for a conventional
fiber. S is comparable to the value for a PBF, and the main contribution
is again the lengthening of the fiber.

Example 3

[0162]FIG. 20 schematically illustrates an example configuration for
testing a fiber optic gyroscope 905 compatible with certain embodiments
described herein. The sensing coil 910 comprises a Blaze Photonics
air-core fiber having a length of 235 meters wound in 16 layers on an
8-centimeter-diameter spool using quadrupole winding to reduce the
thermal and acoustic sensitivities of the coil 910. Each fiber layer in
the coil 910 was bonded to the layer underneath it with a thin epoxy
coating, and the outermost layer was also coated with epoxy. This fiber
was essentially single-moded at the signal wavelength of about 1.54
microns (the few higher order modes were very lossy). The calculated
scale factor of this coil 910 was 0.255 s. Light from a broadband
Er-doped superfluorescent fiber source (SFS) 920 was isolated by isolator
922, transmitted through a polarization controller (PC1) 924, and
split by a 3-dB fiber coupler before being coupled to a fiber-pigtailed
LiNbO3 integrated optical circuit (IOC) 930 comprising a polarizer
932, a 3-dB input-output Y-junction coupler 934, an electro-optic phase
modulator (EO-PM) 936, and a polarization controller PC2 938. The
latter was modulated at the loop proper frequency (600 kHz) with a
peak-to-peak amplitude of 3.6 rad to maximize the FOG sensitivity.

[0163]The Y-coupler 934 splits the input light into two waves that
propagate around the fiber coil 910 in opposite directions. The two ends
of the output pigtails of the IOC 930, cut at an angle to reduce
back-reflections and loss, were butt-coupled to the ends of the PBF
sensing coil 910, which were cleaved at 90°. Measured losses were
about 2-3 dB for each butt-junction, about 4.7 dB for the fiber coil 910,
and about 14 dB round-trip for the IOC 930. The PC.sub./924 was adjusted
to maximize the power entering the interferometer, and the PC2 was
adjusted for maximum return power at the detector (about 10 μW for 20
mW input into the IOC 930). The detector 940 was a low-noise amplified
InGaAs photodiode (available from New Focus of Beckham, Inc. of San Jose,
Calif.). Rotating the coil 910 around its main axis induces a phase shift
via the Sagnac effect between the counter-propagating waves, and the
phase shift is proportional to the rotation rate Ω. The modulated
light signal returning from the coil 910 was detected at the fiber
coupler output and analyzed with a lock-in amplifier 950 (100-ms
integration time; 24-dB/octave filter slope). This measured signal has a
linear dependence on the phase shift, for small rotation rates. The
gyroscope sensitivity was maximized by applying a sinusoidal phase
modulation to the two waves at the loop proper frequency with the EO-PM
936.

[0164]The short-term noise of this air-core fiber gyroscope 905 was
measured by recording the one-sigma noise level in the return signal as a
function of the square root of the detection bandwidth for integration
times ranging from 100 microseconds to 10 seconds. This dependence was
found to be linear, as expected for a white-noise source, with a slope
that gave the gyroscope's random walk. This measurement was repeated for
different signal powers incident on the detector. FIG. 21 shows the
measured dependence of random walk on the signal power, measured at the
IOC input. This result indicates clearly that the sensitivity of the
air-core fiber gyroscope was limited by two of the three main sources of
noise typically present in conventional fiber gyroscopes, namely detector
thermal noise for low detected powers (e.g., less than 4 μW) and
excess noise from the broadband light source for high detected powers
(e.g., greater than 4 μW). The third source of noise is shot noise. At
low power, the minimum detectable rotation rate was inversely
proportional to the detected power, while at higher power the noise was
independent of power. The dashed curve labeled "thermal noise" in FIG. 21
is the theoretical contribution of the detector thermal noise, calculated
from the detector's noise equivalent power (2.5 pW/ Hz). The horizontal
dashed curve represents the predicted excess noise calculated from the
measured bandwidth of the SFS (2.8 THz). The lowest dashed curve
represents the theoretical shot noise which is inversely proportional to
the square root of the input power. This contribution is negligible in
the air-core gyroscope 905. The sum of these three sources of noise is
the total expected noise, illustrated by the solid curve in FIG. 21. It
is in good agreement with the measured data points. This comparison
demonstrates that the performance of this air-core fiber gyroscope is
limited by excess noise, as it typically is in conventional FOGs.
Importantly, it also shows that the residual back-reflections from the
butt-coupling junctions between dissimilar fibers, as schematically
illustrated by FIG. 20, have no impact on the gyroscope's short-term
noise, presumably because of the use of a low-coherence source, and
because the lengths of the two pigtails differ by more than one source
coherence length, which further reduces coherent interaction between the
primary and the reflected waves.

[0165]In the excess-noise-limited regime, the random walk of the air-core
fiber gyroscope is 0.015 deg/ hr. For an integration time of 80
milliseconds, corresponding to a typical detection bandwidth of 1 Hz, the
measured minimum detectable phase shift is then 1.1 μrad,
corresponding to a minimum detectable rotation rate of 0.9 deg/hr. These
values are very similar to the performance of state-of-the-art commercial
inertial-navigation-grade fiber optic gyroscopes. This result was
obtained while using the same detected power as in a typical conventional
FOG (e.g., about 10 μW). However, because of the higher propagation
loss of air-core fibers, this power was achieved by using a larger input
power than a conventional FOG, namely a few mW, as shown by FIG. 21. This
input power could nevertheless easily be reduced by using a lower noise
detector, which would move the crossing point of the two dashed curves of
FIG. 21 to a lower power, and would reduce the fiber loss.

[0166]The benefits of the air-core fiber gyroscope in certain embodiments
lie mainly in its improved long-term stability, starting first with its
temperature drift. A thermal transient applied to a Sagnac loop anywhere
but at its mid-point induces a differential phase shift indistinguishable
from a rotation-induced phase shift. If the temperature time derivative
is {dot over (T)}(z) in an element of fiber length dz located a distance
z from one end of the coiled fiber, the total phase shift error in a
fiber of total length L is:

where λ0 is the wavelength and c is the velocity of light (both
in vacuum), n is the effective index of the fiber mode, and S is the
Shupe constant. The Shupe constant takes into account both the fiber
elongation and the effective index variation with temperature, and is
independent of fiber length. The phase shift error of Equation (21)
induces a rotation-like signal ΩE related to
ΔφE by:

Δφ E = 2 π λ 0 c LD
Ω E ( 22 ) ##EQU00017##

where D is the coil diameter.

[0167]Substituting Equation (21) into Equation (22), and using a
dimensionless variable z'=z/L, yields the following expression for the
rotation rate error induced by the transient temperature change {dot over
(T)}(z):

[0168]Equation (23) states that the thermal sensitivity of the FOG is
proportional not only to the Shupe constant S, but also to n2, the
square of the mode index. Since the air-core fiber has a much smaller
effective index (n≈0.99) than does a standard fiber
(n≈1.44), as well as a smaller Shupe constant, a dramatic
reduction of the thermal sensitivity of the gyroscope is expected by
using an air-core fiber. As shown in Table 3, the Shupe constant for the
SMF28 fiber was measured to be S=7.9 μm/° C. and the Shupe
constant for the Blaze Photonics air-core fiber was measured to be S=2.2
ppm/° C. Combined with the additional benefit of this n2
dependence, these values suggest that the Blaze Photonics PBF gyroscope
should be about 7.6 times less thermally sensitive than the solid-core
fiber gyroscope, which constitutes a considerable stability improvement.

[0169]To verify these predictions experimentally, the output signal of the
air-core FOG and of a solid-core FOG were recorded while subjecting the
coils to known temperature cycles. In each case, the sensing coil was
heated asymmetrically by exposing one of its sides to warm air from a
heat gun, as schematically illustrated by FIG. 20. Prior to these
measurements, each gyroscope was carefully calibrated by placing it on a
rotation table, applying known rotation rates, and measuring the lock-in
output voltage dependence on rotation rate. The quantity measured during
the thermal measurements was therefore a rotation error signal
ΩE from which ΔφE was inferred using Equation
(22).

[0170]FIG. 22A shows an example of a measured temporal profile applied to
one side of the air-core fiber gyroscope coil and the measured rotation
error that it induced. Since the rotation error depends on the time
derivative of the temperature, as expressed by Equation (23), FIG. 22B
illustrates the derivative of the applied temperature change. This
derivative was calculated numerically from the measured temporal profile
of FIG. 22A, then filtered numerically to simulate the 4-stage,
24-dB/octave low-pass filter of the lock-in amplifier. Comparison to the
measured rotation error, reproduced in FIG. 22B, shows a reasonable
agreement between the two curves, in agreement with Equation (23).

[0171]In a quadrupolar winding, as illustrated in FIG. 23, the first
(outermost) layer is a portion of the sensing fiber located close to one
of the two Sagnac loop ends closest to the coupler (e.g., between
positions z=0 and z=L1). The second layer, underneath it, is a
portion of the sensing fiber located at the opposite end of the coil
(Ln-1<z<Ln=L). The third layer is a portion of the
sensing fiber located next to the second layer
(Ln-2<z<Ln-1), etc. Just after the heat has been turned
on, the first layer of the coil heats up first, and as a result, the
differential phase between the counter-propagating waves changes (e.g.,
increases). As heat continues to be applied, it propagates radially into
the coil, and the second layer, then the third layer, start to warm up.
In a quadrupolar or a bipolar winding, the first and second layers are
located symmetrically in the Sagnac loop. Hence as the second layer heats
up, the thermal phase shift it induces begins to cancel that induced in
the first layer. The same cancellation process takes place for the deeper
layers. The total phase shift, however, continues to increase because the
first layer heats up faster than the internal layers. Eventually, the
temperature of the outer layer reaches some maximum value, and as more
internal layers gradually heat up the total thermal phase decreases. If
heat is applied long enough, the temperature along the fiber reaches a
steady-state distribution, and the thermal phase shift vanishes.

[0172]This behavior is consistent with the observed behavior of the
thermally induced signal, which increases first, then decreases over
time, as shown by FIG. 22B. The measured signal closely follows the
temperature derivative for about 1 second after the heat was turned on.
For longer times, the two curves disagree in that the measured rotation
error curve drops below the temperature derivative curve because, by
then, heat has reached deep into the coil and the quadrupolar winding
starts canceling the thermal phase shift. Just after the heat is turned
off (around t=5.5 seconds in FIG. 22B), the rotation error becomes
negative. The reason is that at that time, the outermost layer starts to
cool down. Hence the sign of the temperature gradient is reversed, and so
is the sign of the rotation error.

[0173]When the air-core fiber was replaced by the solid-core fiber coil,
the behavior of the gyroscope was similar, as shown by FIGS. 24A and 24B.
The rotation error increased just after the start of the heat pulse, then
decreased, and finally became negative after the heat was turned off.
However, the solid-core fiber gyroscope was clearly much more sensitive
to asymmetric heating than was the air-core FOG. For comparable applied
peak derivative {dot over (T)} (75.5° C./s for the SMF28 vs.
41.1° C./s for the PBF), the error signal was about 10 times
larger for the solid-core fiber gyroscope, as shown by a comparison of
FIGS. 22B and 24B.

[0174]The measurements provided the temperature derivative {dot over
(T)}(z) at all times but only at the surface of the coil (z=0). It was
consequently not possible to apply Equation (23) and extract from the
measured rotation error signals a value for the Shupe constant S of the
two fiber coils. However, the thermal performance of the two gyroscopes
can still be compared by making two observations. First, because the two
coils have identical diameter and thickness, the rates of heat flow are
expected to be comparable in the two coils. Second, based on the above
discussion regarding the dynamic of heat flow in a quadrupolar coil, the
total thermal phase shift is expected to reach its maximum shortly after
the first layer has started to heat up. The maximum thermally induced
rotation error can therefore be approximated by:

Furthermore, the rate of temperature change of the outermost layer is
expected to be close to the rate of temperature change measured at the
surface of the coil, and to be weakly dependent on z'. Hence, in Equation
(24), {dot over (T)}(z') can be taken out of the integral, which shows
that ΩE.max should scale approximately linearly with the
measured surface temperature derivative.

[0175]To verify this approximation, the dependence of the maximum rotation
rate error on the applied temperature gradient was measured for each
gyroscope. For example, in the measurement shown by FIG. 22B, the maximum
rotation rate error is equal to 0.02 deg/s, and at the time the maximum
rotation rate error occurred (t≈1.8 s), the applied temperature
gradient was about 41.1° C./s. FIG. 25 shows the dependence of the
maximum rotation rate error on the applied temperature gradient measured
in both the conventional solid-core fiber gyroscope and in the air-core
fiber gyroscope. The maximum rotation rate error increases roughly
linearly with applied temperature gradient, which confirms the validity
of the approximation. The slope of these dependencies are
2.4×10-3 deg/s/(° C./s) for the SMF28 fiber gyroscope,
and 2.9×10-4 deg/s/(° C./s) for the air-core fiber
gyroscope. After correcting for the slightly different length L of the
two sensing fibers, for identical coil lengths, the air-core fiber
gyroscope is 6.5 times less sensitive to temperature gradients than the
conventional FOG.

[0176]Independent thermal measurements performed on short pieces of the
same fibers showed that the ratio of Shupe constants for the SMF28 and
the Blaze PBF is 3.6, as shown in Table 3. When using these values in
Equation (23), together with mode effective indices of 0.99 and 1.44 for
the two fibers, respectively, and assuming identical coils, the air-core
fiber is expected to be 7.6 times less sensitive than the solid-core
fiber to thermal perturbations. This value is in good agreement with our
experimental value of 6.5. The small difference (about 13%) may be due to
slightly different heat propagation properties in the coils, which is
expected since an air-core fiber constitutes a better thermal insulator.
The measured value of 6.5 is also in excellent agreement with the
theoretically predicted ratio of 6.6 for these two fibers. In any case,
measured and theoretical values demonstrate unequivocally the significant
advantage of using an air-core fiber in a FOG to reduce its thermal
sensitivity. Further design improvements (e.g., optimization of the
jacket) can result in the Shupe constant being reduced by another factor
of about 3, bringing the total improvement over a conventional coil to a
factor of about 23.

[0177]The second significant long-term stability improvement provided by
an air-core fiber gyroscope is a dramatic reduction in the non-reciprocal
Kerr effect. To illustrate this improvement, the magnitude of the
Kerr-induced drift in the air-core FOG was measured by observing the
change in the gyroscope output when the power between the two
counter-propagating signals was intentionally unbalanced. To be able to
observe this very weak effect, the IOC was replaced by a 10% fiber
coupler, which provided a strong imbalance between the
counter-propagating powers. This change was accompanied by the
replacement of the other components present on the IOC (polarization
filter and phase modulator) by a standard fiber polarizer and a
piezoelectric fiber phase modulator. The SFS, which almost completely
cancels the Kerr effect, was replaced by a narrow-band semiconductor
laser. With the narrow-band source, the noise due to coherent
back-scattering from the fiber was quite large (about 19 dB higher than
with the SFS). In fact, even with the largest input power tolerated by
the optical components (e.g., 50 mW), back-scattering noise exceeded the
Kerr-induced signal. In other words, the Kerr effect of the air-core
fiber gyroscope was too weak to measure. Nevertheless, by recognizing
that the Kerr phase shift was at most equal to the
(back-scattering-dominated) noise, this measurement provides an estimate
of the upper bound value of the fiber mode's Kerr constant. After
correcting for the known Kerr contribution from the (solid-core) fiber
pigtails inside the Sagnac loop, the Kerr constant was found to be
reduced by at least a factor of 50 compared to the same gyroscope using
the SMF28-fiber coil. This result confirms that the Kerr effect is
substantially reduced in an air-core fiber, by a factor of 50 or more.

[0178]While the effect of a magnetic field on the air-core fiber gyroscope
was not measured directly, the Verdet (Faraday) constant of a short
length of the air-core fiber was measured. These measurements indicate
that for equal length, the Faraday rotation induced by an applied
magnetic field is about 160 times weaker in the air-core fiber than in an
SMF28 fiber. Inferring an accurate value of the Verdet constant of the
air-core fiber from this result utilizes a precise knowledge of the
fiber's birefringence. This constant is estimated to be at least a factor
of about 10 dB smaller than that of an SMF28 fiber, and it could be as
low as about 26 dB smaller than that of an SMF28 fiber. In practice, an
air-core FOG requires much less pt-metal shielding (if any) than do
current commercial FOGs, which will reduce the size, weight, and cost of
the air-core FOG as compared to conventional FOGs.

[0179]In certain embodiments described herein, the temperature dependence
of an FOG is advantageously reduced by using an air-core fiber. In
certain other embodiments, the temperature dependence of other types of
interferometric fiber sensors can also be advantageously reduced. Such
fiber sensors include, but are not limited to, sensors based on optical
interferometers such as Mach-Zehnder interferometers, Michelson
interferometers, Fabry-Perot interferometers, ring interferometers, fiber
Bragg gratings, long-period fiber Bragg gratings, and Fox-Smith
interferometers. In certain embodiments in which the fiber sensor
utilizes a relatively short length of air-core fiber, the additional
costs of such fibers are less of an issue in producing these improved
fiber sensors.

[0180]In certain embodiments described herein, the temperature dependence
of an FOG is advantageously reduced by using a Bragg fiber. In certain
other embodiments, the temperature dependence of other types of
interferometric fiber sensors can also be advantageously reduced. Such
fiber sensors include, but are not limited to, sensors based on optical
interferometers such as Mach-Zehnder interferometers, Michelson
interferometers, Fabry-Perot interferometers, ring interferometers, fiber
Bragg gratings, long-period fiber Bragg gratings, and Fox-Smith
interferometers. In certain embodiments in which the fiber sensor
utilizes a relatively short length of Bragg fiber, the additional costs
of such fibers are less of an issue in producing these improved fiber
sensors.

[0181]While certain embodiments have been described herein as having a
photonic-bandgap fiber with a triangular pattern of holes in the
cladding, other embodiments can utilize a photonic-bandgap fiber having
an arrangement of cladding holes that is different from triangular,
provided that the fiber still supports a bandgap and the introduction of
a hollow core defect supports one or more core-guided modes localized
within this defect. For example, such conditions are satisfied by a fiber
with a cladding having a so-called Kagome lattice, as described in "Large
pitch kagome-structured hollow-core photonic crystal fiber," by F. Couny
et al., Optics Letters, Vol. 31, No. 34, pp. 3574-3576 (December 2006).
In certain other embodiments, a hollow-core fiber is utilized which do
not exhibit a bandgap but still transmit light confined largely in the
core over appreciable distances (e.g., millimeters and larger).

[0182]Those skilled in the art will appreciate that the methods and
designs described above have additional applications and that the
relevant applications are not limited to those specifically recited
above. Moreover, the present invention may be embodied in other specific
forms without departing from the essential characteristics as described
herein. The embodiments described above are to be considered in all
respects as illustrative only and not restrictive in any manner.